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Calculus 1 / AB
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August 10, 2021
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August 10 of 2021
Suppose a baseball is thrown vertically upward from the surface of a planet. Its height (in meters) after t seconds is s(t) = (30)t - (0.357)t^2. Use calculus to find: How many seconds will it take for the ball to reach the maximum height?
3. A cake shop needs to make boxes in a cylinder shape. Each box has to have a volume of 10π m3. The materials for the top and bottom of the box costs 6$/m2 and materials for the side of the box costs 4$/m2. Find the radius of the box that will minimize the building costs.
2-Pure water, 4 times its volume, is added to the 30% alcohol solution. What is the new alcohol percentage? 4-There are 60g of A, 30g of B, and 20g of substance C in the 110g X blend. How many grams of X mix can be made with 360g of A, 400g of B, and 500g of substance C? 6-6 apprentices do 3…
a) Find the local maximum and minimum values and saddle points of the function f(x,y) = x^2y^2 - 8x + y/xy. Graph the factors (roughly) with a domain of view point that reveal all the important aspects of the function. f_x = y - x^(-2) = 0 f_y = x + 8y^(-2) = 0 b) A cardboard box without a…
According to a study, the probability that a randomly selected teenager studied at least once during the week was only 0.54. Let X be the number of teenagers who studied at least once during the week. What is the probability that only 4 students from your study group of ten have studied this…
stone height: 112 tokens <thought
PROMPT: (Problem, PART ONE): A motorboat travels 80 miles in 2 hours going upstream. It travels 112 moles going downstream in the same amount of time. What is the rate of the boat in still water and what is the rate of the current? RATE IN THE BOAT IN STILL WATER: ____ mi/h RATE OF THE CURRENT:…
a) A printed page is to have 60 cm² of printed material with 5 cm margins on the sides and 3 cm margins on top and bottom. What is the minimum total area of the page? b) An open-topped storage box with a square base is to have a capacity of 5 m³. Sides cost $1.60/m² while material for the…
1. Find ∂f/∂x where f(x, y) = sin(x/y). (a) (x/y) cos(x/y). (b) (-x/y)^2 cos(x/y). (c) ((y - x)/y)^2 cos(x/y). (d) (1/y) cos(x/y). (e) None of the other choices. 2. Find ∂f/∂y where f(x, y) = sin(x/y). (a) (x/y) cos(x/y). (b) (-x/y^2) cos(x/y). (c) ((y - x)/y^2) cos(x/y). (d) (1/y)…
A cable runs along the wall from C to P at a cost of $5 per meter, and straight from P to M at a cost of $13 per meter. Let x be the distance from C to P. If M is 24 meters from the nearest point A on the wall where P lies, and A is 53 meters from C, find x such that the cost of installing the…
Consider the following statement: if for every p ∈ Q, there exists q ∈ Q such that |p - q| < 1, then 2^10 ≤ 10^3. (a: 4 points) Write down the negation of the boldface statement. (The boldface statement contains English words; so should your answer.) (b: 4 points) Write down the converse…
Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the…
Suppose the distance s (in feet) covered by a car moving along a straight road after t seconds is given by the function s = f(t) = 3t^2 + 13t. Calculate the instantaneous velocity of the car when t = 35 s. Group of answer choices: 4130 223 350 445 2. If a rock is thrown upward on the planet…
Find the x-values where points of inflection occur. State the x-intervals where the function is concave up and the x-intervals where it is concave down. f(x) = -x^3 + 9x^2 - 55 What are the x-values of the inflection point(s)? A. The x-values of the inflection point(s) are x = (Use a comma…
Find the center of mass of the given system of point masses lying on the x-axis. m1 = 6, m2 = 3, m3 = 5 x1 = -5, x2 = 0, x3 = 3
A gun with a muzzle velocity of 600 m/s is fired at an angle of 12° above the horizontal. Find the horizontal and vertical components of the velocity.
A study of a country's colleges and universities resulted in the demand equation q = 20,000 - 2p, where q is the enrollment at a public college or university and p is the average annual tuition (plus fees) it charges. Officials at Enormous State University have developed a policy whereby the…
A rock is dropped off of a tall building. Its height (in feet) after t seconds is given by s(t) = -16t^2 + 1000. a. What is its velocity 1 second after being dropped? Include units. Show work. b. What is its acceleration 1 second after being dropped? Include units. Show work.
Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x) = 2x^4 - 2x^3 + 5 Determine the intervals on which the given function is concave up or concave down. Select the correct choice below and fill in the answer box(es) to…
A continuous function f, defined for all x, has the following properties: 1. f is decreasing. 2. f is concave up. 3. f(13) = -3. 4. f'(13) = -14. Sketch a possible graph for f and use it to answer the following questions about f. A. For each of the following intervals, what is the minimum and…
The graphs of f and g are given in the figure. There are two curves graphed on the xy coordinate plane. The first curve, labeled f, begins at the point (-4, -2), goes up and right passing through the points (-3, -1) and (-2, 1), reaches a maximum point on the y-axis at y = 3, goes down and…
Victor is on a mountain whose height at the location (x, y) is given by z = x(6 + sin y). If Victor is at (1, π/2), in what direction should he take to get down the mountain as fast as possible?
Complete the following steps for the given integral and the given value of n. a. Graph the integrand on the interval of integration. b. Calculate Δx and the grid points x0, x1,...,xn assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d.…
The number of Americans age 65 or older can be modeled as n(x) = -0.00082x^3 + 0.059x^2 + 0.183x + 34.42, where output is measured in million people and x is the number of years since 2000, projections through 2050.†(a) What is the projected number of Americans 65 years of age and older in…
In the diagram shown, a puck A of mass 0,024 kg is moving to the right along the x-axis with a speed of 6.08 m/s. It collides with puck B of mass 0.048 kg initially at rest. The collision is not head-on. After the collision, the two pucks fly apart in the directions shown in the…
1. Find the length of the curve y = , 0 ≤ x ≤ 1 Length of curve = 2. The mean value of a function is defined on the interval [a , b] as Find the mean value of f(x) = on the interval [0,].
Suppose you are on a ferris wheel with a diameter of 20 m and making one turn per minute. How fast do you rise or fall when you're 6 m horizontally from the vertical line that runs through the center of the wheel?
A car traveled 50 miles due south then traveled 90 miles 35 degrees south of west. Solve for the resultant displacement and its direction.
In your company, the production costs in SRs per week of producing x units is given by C(x) = 3x + x^2 + 6x^3 and the demand function for the units is given by p(x) = x - x^2. What is the marginal cost function, marginal revenue function, and marginal profit function?
Approximate the area under f(x) = x^3 - 8x^2 + 15x + 10 from x = 1 to x = 5 using (a) 6 rectangles and starting with the right endpoint (R6) (b) 3 rectangles and starting with the left endpoint (L3) (c) Compare both of your approximations with the exact area under the curve and determine if…
Find the critical points of the function f(x) = 5x^2 - 10x. Apply Rolle's theorem for the function f(x) = x^2 + 5x over the interval [-5,0]. Justify your answer. Verify the Mean Value Theorem for the function f(x) = 2x^3 + x + 5 over the interval [0,1]. A ball is dropped from a certain height.…
A model rocket is vertically launched and its position is modeled by s(t) = -5t^2 + 10(2)(5)t + 3, where s is measured in meters and t is in seconds. Determine the equations of velocity and acceleration. b) Determine its initial position, initial velocity, and initial acceleration. c) Determine…
Module Six Discussion Question: Solve the problems below. Copy the description of your Ferris wheel in the text box and include that as part of your initial Discussion post in Brightspace. Using "copy" from here in Mobius and "paste" into Brightspace should work. Hint: This is similar to…
A particle moves in a straight line with the given velocity ð‘£(ð‘¡) = 3ð‘¡Â² - 30ð‘¡ + 27 (in m/s). Find the displacement and distance traveled over the time interval [0,10]. (Give your answers as whole or exact numbers.) Displacement: Total distance traveled:
A car travels along a straight road for 30 seconds starting at time t = 0. Its acceleration in ft/sec^2 is given by the linear graph below for the time interval [0, 30]. At t = 0, the velocity of the car is 0 and its position is 10. What is the total distance the car travels in this 30-second…
Suppose a small (radius zero) ball is rolling along the curve 𑦠= 18/x³ + 1 with a constant speed of 3 ð‘š/ð‘ . What is the velocity vector of the ball when it passes through the point (2,2)?
A linear cost function is C(x) = 7x + 700 (Assume C is measured in dollars.) (a) What are the slope and the C-intercept? slope: 7 C-intercept: 700 (b) What is the marginal cost MC? MC = 7 What does the marginal cost mean? a) Each additional unit produced reduces the cost by this much (in…
Suppose that a survey of 100 advertisers in U.S. News, These Times, and World found the following: - 15 advertised in all three - 28 advertised in These Times and U.S. News - 31 advertised in World and U.S. News - 29 advertised in World and These Times - 59 advertised in These Times - 55…
If an arrow is shot straight upward on the moon with a velocity of 66 m/s, its height (in meters) after t seconds is given by s(t) = 66t - 0.83t^2. What is the velocity of the arrow (include units) after 7 seconds? How long will it take for the arrow to return and hit the moon (include…
Consider the curve c(t) = <t, sin(3t), cos(3t)>. a. Compute the velocity and acceleration. b. Find the length of the curve over the interval [0, π]. c. Find the curvature κ.
The market demand curve for a duopoly is P = 10 - 4Q. Fill in the entries for each of the following duopoly models (Please show your calculation). Marginal cost for both firms is $26. Using the Collusion model: Model Q1 Q2 Q P Profit margin 1 Profit margin 2 Collusion
Consider the following given function and given interval. g(x) = (x + 2)^3, [0, 2] (a) Find the average value gave of g on the given interval. gave = (b) Find c in the given interval such that gave = g(c). (Enter your answers as a comma-separated list. Round your answers to three decimal…
An object is propelled vertically upward from the ground with an initial velocity of 43.4 m/s. The signed distance s (in meters) of the object from the ground after t seconds is given by the position function s = s(t) = -4.9t^2 + 43.4t. (a) What is the velocity, in meters per second, of the…
A kite 50 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 150 ft of string has been let out? Answer (in radians per second):
Baggzz, Inc. sells custom backpacks to the students of a community college in California's San Joaquin Valley. It determines that in order to sell x backpacks, the price per backpack must be p = 150 - 0.75x. It also determines that the total cost of producing x backpacks is given by C(x) = 3000…
A point has horizontal and vertical displacements (in cm) of x = t3 - t2 + 5t and y = 24 - t3 + t. Find the magnitude and direction of the resultant velocity at t = 3.50 s.
a kite is flying 40 ft above the ground at the end of a string. the girl flying the kite lets out the string at a rate of 1 ft per second. If the kite remains 40 ft above the ground How fast the horizontal distance from the girl to the point below it increasing when 50 ft string is out
The cost of producing x teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula C(x) = 100 + 39x - 0.06x^2. (a) Find the marginal cost function C'(x). C'(x) = Use it to determine how fast the cost is going up (in $) at a production…
(2) Deductive reasoning skill: Deductive reasoning is a logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true. Please watch the video clip provided on Deductive Reasoning. Do the following problems (Q1 - Q5) and then answer…
A projectile if fired with an initial speed at 100m/s and an angle of elevation of 45 degrees. A) find the position vector function, r(t), for the projectile. B) Find the range of the projectile. C) Find the maximum height reached.
A piece of wire 13 m long is cut into two pieces. One piece is bent into the shape of a circle of radius ð‘Ÿ and the other is bent into a square of side ð‘ . How should the wire be cut so that the total area enclosed is: a) a maximum? ð‘Ÿ= and ð‘ = . b) a minimum? ð‘Ÿ= and ð‘ =.
1a) A power house, P, is on one bank of a straight river 200 m wide, and a factory, F, is on the other bank 400 m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen so the cost is…
Consider the following cost and revenue functions for a city pair route that an airline is the single seller for: C(q) = 0.001q^3 - 1.8q^2 + 1180q + 134000 R(q) = -0.859q^2 + 1374.4q Required: A) Develop a profit function B) At what value of q are profits at a maximum? C) What is the revenue…
The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The marketing department has determined that the demand for these speakers is represented by the following function, where p denotes the speaker's unit price (in dollars) and x denotes the quantity…
1a.) Suppose that an object moves along an s-axis so tha its location is given by s(t)= t^2 + 5t at t. (Here s is in meters and t in seconds) Find the average velocity of the objects in meters per second over the time interval t=2 to t=8 seconds. 1b.) Find the instantaneous velocity of the…
1. Find the slope of the tangent line to the curve (√3x+√y)+√6xy=9 at the point (2,3). The slope of the tangent line to the curve at the given point is _______. 2. Find dx/dy by implicit differentiation when xy^3/(sec(y)−2)=1+y^4. dy/dx = _______. 3. Consider the function…
Find a cubic polynomial f(x) = ax^3 + bx^2 + cx + d for the landing of a fighter jet. The function f(x) will be the height of the plane from the ground, and x the ground distance of the plane away from the airport. Everything is measured in miles. The fighter jet is cruising at 35,000 feet and…
Distance A ball is dropped from a height of 9.8 meters. Each time it drops h meters, it rebounds 0.81h meters. Find the total distance traveled by the ball.
A rock is dropped from the edge of a cliff that is 320 meters above the water. What is its velocity at impact? (Hint: Use the height function h(t) = 5t^2 + v0t + s0 for free-falling objects.)
II. We are interested in trying to predict the expenses of a men's college/university soccer team using two explanatory variables: the expenses of the men's golf team and the expenses of the men's baseball team. A random sample of 200 colleges and universities was taken from institutions with a…
You are stationary on the ground and are watching a bird fly horizontally towards you at a rate of -6 m/s. The bird is located 60 m above your head. How fast does the angle of elevation of your head change when the horizontal distance between you and the bird is 160 m? (Leave your answer as an…
a) State the Mean Value Theorem. b) Sketch a graph that illustrates the conclusion of the Mean Value Theorem.
A projectile is shot upward from the surface of Earth with an initial velocity of 111 meters per second. Use the position function below for free-falling objects. What is its velocity after 2 seconds? After 14 seconds? (Round your answers to one decimal place.) s(t) = -4.9t^2 + v0t + s0 s =…
The government is trying to get more teenagers in juvenile detention to go to school while they are in detention. The management at the facility wants to test if cash payments will increase school attendance. They randomly split the children (PP) into a treatment and control group (PP) where…
a) A manufacturer estimates that the hourly cost of producing x units of a product on an assembly line is C(x) = 0.2x^3 - 8x^2 + 146x + 300 dollars. b) What is the change in cost when production changes from 10 units to 50 units? Include units. c) What is the average rate of change of cost when…
5. Find the area bounded by the graphs of the indicated equations over the given interval. y = e^x; y = -1/x; 1 ≤ x ≤ 2 The area is ? square units. 6. Find the consumers' surplus and the producers' surplus at the equilibrium price level for the given price-demand and price-supply…
Find the intervals on which the function is continuous. 6) y = 4√(10x - 10) A) (-∞, 1) B) (1, ∞) C) (1, ∞) D) (-∞, 1) Find dy/dx. 7) s = t^3 - csc(t) + 17 A) 3t^2 - csc(t) cot(t) B) t^2 - cot^2(t) + 17 C) 3t^2 + cot^2(t) D) 3t^2 + csc(t) cot(t) Find dy/dx by implicit…
A rock is thrown from a Martian cliff which has a height of 288 feet. The initial velocity of the rock is 9 feet per second. Its speed in feet per second at time t seconds after it is thrown is given by the equation: s(t) = 3t + 9 How far is the rock from the ground 3 seconds after it is…
A baseball diamond is a square with sides measuring 90 ft. A batter hits the ball and runs toward first base with a speed of 29 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the…
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = 5x^2 - 4x + 1, [0, 2] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [0, 2] and differentiable on (0, 2) since…
A baseball player at second base throws a ball 90 feet to a player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 50 miles per hour and at an angle of 15 degrees above the horizontal. At what height does the player at first base catch the ball?
Draw a Markov chain with at least 4 states, and then write the matching transition matrix. Be sure to explain what the situation is that your Markov chain is modeling. What do each of the states mean? Why do the probabilities make sense?
Consider the function ln x. a. Normalize it so that it is a PDF on [2, 5]. b. Compute the expected value of the PDF on that interval. c. Compute the variance of the PDF on that same interval.
20.) At a price of $60 for a 4-hour city tour, a bus tour company attracts 120 customers. For every $2 decrease in price, an additional four customers are attracted. The cost, in dollars, for accommodating ð‘¥ customers on the tour is: ð¶(ð‘¥) = 0.1ð‘¥^2 + 12ð‘¥ + 1500 a. Find the demand…
A lamina has the shape of the planar region bounded by y = x^2, y = 0, x = 0, and x = 1. The density at each point of the lamina is given by p(x, y) = x^2 + 2y. Find the mass m of the lamina and the x-coordinate x* of the center of mass of the lamina.
Let f be the function given by f (x) = x3 – 6x2 . What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [0, 6]. ( A ) 0 only ( B ) 4 only ( C ) 0 and 4 (D ) 4 and 6 ( E ) None of these
Approximate the area under the curve for the function f(x) = 9x^3 from x = 0 to x = 3 by summing the areas of 'thin' rectangles. a) Use n = 3 rectangles (of the same width). a1) Find a left-endpoint sum by using the left endpoint for the height of each rectangle. a2) Find a right-endpoint sum…
An object is launched at an angle θ = π/4 from the horizontal with an initial speed of 80 m/s. Determine the time of flight and the horizontal range. The object is launched from the ground, and air resistance is ignored.
A gun is fired at an angle of 30 degrees above horizontal. What is the muzzle speed (initial speed) of the bullet if the maximum height attained by the bullet is 500 meters?
The owner of the ranch wants to fence off a rectangular piece of land of area 1800 yd². He intends to use an existing wall as one of the sides. The fence cost $10 per yard. What should the dimensions be in order to minimize the total cost of the fence? What is the total cost?
A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure. Let r represent the radius of the circle and h the height of the rectangle. To ensure that the window admits enough light, the area of the window must be 90 square feet. (a) Write a formula…
As shown below, an object weighing 300 N is lying on an inclined plane. 45° (a) Write the force of gravity as a vector in 2-space. (b) Use projections to find the vector component of the force of gravity acting parallel to the ramp. Do not rotate the axes! (c) If the object moves 40 m down the…
Given r = f(theta) = 1 + sin(theta). (15 Points) (a) Graph the curve. (b) Find the area bounded by the curve. (c) Find the points on the curve where the tangent line is horizontal.
Professor Nimrod throws a piece of chalk, at an angle of 39° above the horizon, with initial velocity of 28 feet per second. Given that the chalk was released from a point 6 feet above the ground, find the parametric equations that model the trajectory of the chalk. Also, there is a wall 10…
1. a. Each side of a square is increasing at a rate of 2 cm/s. At what rate (in cm2/s) is the area of the square increasing when the area of the square is 16 cm2? b. The length of a rectangle is increasing at a rate of 7 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 9…
18. what is the maximum area that a rectangle inscribed in a semicircle of radius 13_ / 2 can have? a.13 square centimeters b. 169 square centimeters c. 26 square centimeters d. 338 square centimeters e. 676 square centimeters
(10 points) Suppose that X and Y have the following joint probability distribution: f(x, y) x 0 1 2 0 y 1 2 3/28 9/28 3/28 3/14 3/14 0 1/28 0 0 Evaluate the marginal…
Suppose R(x) dollars gives the revenue from the sale of Clemson hats, when x hats are produced and sold. a). Write a sentence of interpretation explaining that the marginal revenue is 5.28 dollars when 40 hats are sold. (don't use the word "marginal") "When 40 Clemson hats are produced and…
A. If an arrow is shot straight upward on the moon with a velocity of 67 m/s, its height (in meters) after t seconds is given by s(t) = 67t - 0.83t^2. What is the velocity of the arrow (include units) after 5 seconds? B. How long will it take for the arrow to return and hit the moon (include…
Companies produce millions of cans per year to hold their products. For many, they would like to use a can with the minimum surface area to minimize the cost of the can. In this activity, you will determine if a manufacturer's can has been minimized. We first interpret this to mean that the…
Alex needs to buy building supplies for his new patio. He needs 20 bags of cement, 50 cubic feet of sand, and 100 red bricks. There are two building supply stores in town, Rocko's and Big Mike's. The prices for each of the items are shown in the table. Cement: - Rocko's: $4.00 per bag - Big…
A ball is thrown straight down from the top of a 493-foot building with an initial velocity of -30. What is the velocity after 3 seconds? What is the velocity when falling 376 feet?
You go grocery shopping. The price of three apples is 30MU. The price of two oranges and one banana is 18MU. If you return one faulty apple and buy two bananas in total, you are charged 2MU. a) (1 Point) Write down the information as a linear system of equations in matrix vector notation. b)…
The marginal revenue function of the company is G'(x) = -0.4x + 280, in dollars per unit, when x units are sold. 1- Determine the G(x) revenue function of the company. 2- Find the revenue attained by the company when the first 60 units are sold. 3- Find the additional revenue realized by the…
Suppose a stone is thrown vertically upward from the edge of a cliff with an initial velocity of 96 ft/s from a height of 10 ft above the ground. The height s (in ft) of the stone above the ground t seconds after it is thrown is s = -16t^2 + 96t + 10. a) Determine the velocity v of the stone…
a ball weighing 150 newtons lies on an inclined ramp of height 50 m and base 110 m. if the ball slides 15 m down the ramp, calculate the work done in Nm by the force of gravity acting on the ball.
5. Person A earns $40 per hour and Person B earns $42 per hour. ln 40 = 3.689 and ln 42 = 3.738 a. Find the exact percentage by which Person B's hourly salary exceeds Person A's. b. Using logarithms, find the approximate percentage by which Person B's salary exceeds Person A's. 6. Fill in the…
USE CALCULUS, not just physics trig!!! Scientists discover a new planet, Zed. On Zed, the acceleration due to gravity is 32 m/s^2. A bullet is shot from the ground on Zed, at an angle of pi/6 from the ground. The bullet is shot at a velocity of 800 m/s. USING CALCULUS a) derive an equation for…
(6 points) The table below gives the population (in thousands of residents) of a coastal community for the period from 1990 to 1999. Figures shown are for January 1 in each year. Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Population 618 841 1322 1550 1621 1487 957 801 768 738 (a).…
A rectangle is constructed with its base on the diameter of a semicircle with radius 25, and with its two other vertices on the semicircle. Let x be the length (base) of the rectangle that lies on the diameter of the semicircle. Find the dimensions of the rectangle with maximum area. Draw the…
The demand curve for a product is given by q = 144,000 - 600p and the supply curve is given by q = 1000p for 0 ≤ q ≤ 144,000, where price is in dollars. (a) At a price of $100, what quantity are consumers willing to buy and what quantity are the producers willing to supply? (b) Will the…
a kite is flying 40 ft above the ground at the of a string. the girl flying the kite lets out the string at a rate of 1 ft per second. if the kite remains 40 ft above the ground. find the following equation or model if the length of the string is denoted by x and its horizontal distance from to…
A stone is thrown vertically upward from the top of a building 64 ft above the ground with an initial velocity of 48 ft/sec. Ignore air resistance. Determine how long it takes for the stone to strike the ground and with what velocity it will strike the ground.
A cannonball is shot from a cannon with an initial velocity of 24 m/s at an angle of 40° to the ground. a) Determine the initial horizontal and vertical velocity of the cannonball. b) When will the cannonball reach its maximum height? c) How far, measuring along the ground, will it be from its…
1. Use algorithm of curve sketching to sketch the graph of f(x) = x4 + 4x3. 2. A mass of 5 kg is being suspended by two strings, 24cm and 32cm, from two points that are 40cm apart at the same level. Determine the tension in each of the strings.
2. The cost, C, in dollars, of making x large combo pizzas at a local pizzeria can be modeled by the function C(x) = -0.001x³ + 0.025x² + 4x, and the price per large combo pizza is $17.50. Determine A) Revenue function, R(x) b) Marginal revenue function, R′(x) c) Profit function, P(x) d)…
A farmer plans to enclose a rectangular pasture adjacent to a river (see figure). The pasture must contain 5,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing? x=_______m y=_______m
The figure shows a surface z = 2(x^2 + y^2) and a rectangle R in the xy-plane. z = 2(x^2 + y^2) The xy-coordinate plane is given. A surface and a rectangle are given in the first octant. The surface labeled z = 2(x^2 + y^2) starts at the origin and goes up in every direction. The rectangle R…
Find the sum of the maximum and minimum values of the function f(x,y,z) = 6x + 12y + 18z on the ellipse that is formed by the intersection of the cylinder x^2 + y^2 = 36 and the plane y + z = 6.
Properties of Probability, Permutations, Combinations Where applicable, show the permutations/combinations you set up to get the answer. The superintendent of a certain school district has estimated the probabilities associated with the SAT verbal scores of students from that district. The…
Problem three: Let f(0) = 0 = g(1), f(1) = 2 = g(0), and f'(0) = 1, g'(1) = 2. f(t) - g(t + 1) (a) Evaluate: lim (b) Find the derivative of (fog)(r) at r = 1. Problem four: Find dr for see (ry) = r² + y? - 13. Problem five: Consider the function f(x) = cos(2r) + 2x. re [0, T]. (a) Find…
Answer the following questions for the function f(x)=−2x3−3x2+72x−10. a. Find formulas for f′(x) and f′′(x). f′(x)= ____ f′′(x)= ____ Enter f(x), f′(x), and f′′(x) into your grapher to examine the table. b. The formula for the first derivative f′(x) can be factored. Set f′(x)=0 to find the two…
A rock is dropped from a height of 89.5 m and falls toward Earth in a straight line. In t seconds the rock falls 4.9t2 meters. What is the velocity v of the rock when it hits the ground? (Round your answer to two decimal places.)
An analyst for an app determined that the rate of change in the number of app users N(t) over time t (in days) in the first 60 days following a launch can be modeled by: r(t) = 0.06t^2 - 4t + 50 (A) What is the net change in the number of users in the first 30 days? In the next 30 days?…
A gun has a muzzle speed of 70 meters per second. What angle of elevation should be used to hit an object 160 meters away? Neglect air resistance and use g=9.8m/s^2 as the acceleration of gravity.
A ball is thrown at an angle of 60 ∘ to the ground. If the ball reaches the maximum height in 2 seconds, what was the initial speed of of the ball? How far does the ball fly?
There is a jet which is flying in the north direction at a speed of 800 km/h. It then faces a wind blowing in the southeast direction. The wind has a speed of 90 km/h. Calculate the resultant velocity of the airplane. Also, draw a diagram to support your explanation.
f(x) = -x^3 + 3x^2 - 4 1.b. What is the y-intercept of f(x)? 1.c. What are the x-intercepts of f(x)? Hint: (x + 1) is a factor, use synthetic division to find the other factors. 1.d. Find the relative extrema of f(x). State your answers as a coordinate point or "none". Relative Minimum:…
a. A car travels at 𑣠mph. How long does it take to go 120 miles? b. A car travels for 𑡠hours at 𑣠mph. How far does it go? c. A car travels 𑥠miles at 𑣠mph. How long does it take? d. A car travels 𑦠miles in 𑡠hours. What is its speed?
1. A contractor has installed a silt fence around an area that is semi-circular and level to prevent soil from the construction site entering nearby streams. The diameter of the semi-circle is 1,650 feet. How many linear feet of fence does the contractor need to install to enclose the area? 2.…
2) f(x,y) = 5 over the rectangle R = {(x,y): 2 ≤ x ≤ 3, 2 ≤ y ≤ 4} Value of the integral of the constant function is? a) 20 b) 25 c) 50 d) 10 3) f(x,y) = x^2 + y^2 over the unit circle. How to represent the integral form?
Suppose f is continuous on a closed interval [a, b]. If u is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = u. A. Let g(x) = 1/x. Choose an interval where the Intermediate Value Theorem does not apply and explain why. Then, choose a u value for…
8. The height, y, of a ball that is tossed straight up is given by the following formula: -16t^2 + 35t + 15, where the height is measured in feet and t is the time since release measured in seconds. (Negative represents the downward direction.) Include units on all answers. A. At what height…
1. The cost function C(x) = 15,000 + 30x + 1,000/x gives the cost in dollars to produce x units of a commodity. a. Find the marginal cost when the production level is x = 100 units. When they produce 100 units, the marginal costs is ____________ In other words, the approximate cost to produce…
3. Consider the function below. f(x)= (x^2)/(x^2 - 16) (a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = ? Find the horizontal asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist,…
The marginal cost of producing a certain type of light bulb at a production level of x boxes of bulbs is 6 + 0.008x^2 dollars per box of bulbs. Determine the total change in cost if production is increasing from 12 to 149 boxes of bulbs. Total change in cost (in dollars) = ______ 2.…
PLEASE SHOW ALL WORK! A campground owner plans to enclose a rectangular field adjacent to a river. The owner wants the field to contain 125,000 square meters. No fencing is required along the river. a. Let F be the total amount of fencing needed. Given that x and y are the lengths of the sides…
Find the value of k such that the function f(x) = kx^3 is a probability density function on the interval [1,∞). Group of answer choices: a. 3 b. 4 c. 2 d. 1 2. The function f(x) = 1/(4√x) is a probability density function on the interval [1,9]. Find the following (round to 4 decimal places, if…
1. Sketch the graph of a function f that is continuous except that it has a removable discontinuity at x = 3 and a jump discontinuity at x = 5. Explain your answer. 2. If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H =…
A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial…
11. Determine the absolute minimum of f(x) = -5x^3 - 12x^2 + 39x + 6 on the interval [-1,2] a) 4 b) -28 c) -40 d) -48 15. Find the area on the x-axis, under the curve of f(x) = 3sec^2(2x/5) - 7 between x = 1 and x = 2 Select one: a. 10.57 b. 10.96 c. 11.37 d. 11.55 17. Estimate the area under…
Texts: Two forces F1 and F2 with magnitudes 16 lb and 14 lb, respectively, act on an object at a point P as shown in the figure. (Give your answers correct to one decimal place.) (a) Find the resultant force F acting on P. i + j (b) Find the magnitude of F and its direction. |F| = lb 𜃠= °
The random variable x has the following probability distribution: x 1 2 3 4 5 P(x) 2/10 2/10 2/10 1/10 3/10 a.) What is the expected value E(x)? b.) What is the variance V(x)? c.) What is the standard deviation σ?
Two forces of magnitude 80N and 45N act on an object, with an angle between the forces of 100 degrees . What is the resultant force?
A company plans to enclose three parallel rectangular areas for sorting returned goods. The three areas are within one large rectangular area and 992 yd of fencing is available. What is the largest total area that can be enclosed? A rectangle has 2 horizontal sides and 2 vertical sides. Two…
You are shooting a tennis ball, straight up, off the top of a 36 foot high platform using a tennis ball cannon. The cannon can shoot the ball at 36 feet per second. Answer the following questions: (a) How fast is the ball traveling at highest point? (b) At what time is the ball at its peak…
a) An academician wants to see significant mean differences in the performance level of private and public banks in Kabul, Afghanistan. You have been hired as a consultant to suggest a specific quantitative test. Which test would you propose and why? b) In another situation, a manager wants to…
Texts: Two IB schools, A and B, follow the IB Diploma Programme but have different teaching methods. A research group tested whether the different teaching methods lead to a similar final result. For the test, a group of eight students was randomly selected from each school. Both samples were…
Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 2 (a) Find dy/dt, given x = 2 and dx/dt = 14. dy/dt = (b) Find dx/dt, given x = 1 and dy/dt = –7. dx/dt =
1. Locate the centroid of the area bounded by the x-axis and y = -5 + 3x^2. a. (0, -9/5) b. (0, -5/2) c. (0, -3) d. (0, -2) 2. Find the 10th term in the expansion of (b - 3c)^12. a. 1,299,078b^2c^10 b. -1,299,078b^3c^9 c.…
A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plant will germinate, a random sample of 3000…
The Save the World club at a local university is looking to raise money through a bake sale outside of the student center. Out of a desire for efficient fundraising, they consult with economics majors in the club about the best bake-sale strategies. The econ majors do some reading on behavioral…
I conducted a survey among my 58 students in my evening statistics online class to see who eats spaghetti at least twice a week. Based on the frequency, I find that 40% of them eats spaghetti at least twice a week. Therefore, I make a same conclusion for all students from my university. Assume…
Suppose that: f(x)= (4)/(x^2-36) (A) Find all critical values of f, compute their average, and enter it below. Note: If there are no critical values, enter -1000. (B) Use interval notation to indicate where f(x) is increasing. (C) Use interval notation to indicate where f(x) is decreasing. (D)…
Text: 3 Outliers In many statistical analysis programs, data values that are considerably outside the range of the majority of the values are simply dropped from consideration. Using this information, write a program named outliers.py that accepts a number of floating-point values from the user…
An oil company wishes to construct a communications cable from its offshore oil well drilling platform facility A to its refinery B. The offshore facility is 2 miles from the shore, measured perpendicular to the shoreline, and the refinery is 1 miles inland, measured perpendicular to the…
A cannonball is launched at 32 degrees with a velocity of 130 m/s. Write vector equations for the trajectory, the velocity, and the acceleration, and determine the flight time, maximum height, and range for a ground-to-ground shot.
2. Consider the function g(x) = 1 - 1/x + 1/x^2. (a) Find the domain and x, y-intercepts. (b) Find the asymptotes (if they exist). (c) Find the intervals of increasing/decreasing and local extrema. (d) Find the intervals of concavity and inflection points. (e) Sketch the curve. (Note: You may…
2. Find the absolute extrema, if they exist, as well as where they occur for ð‘“(ð‘¥) = 2ð‘¥ + 8ð‘¥^2 + 1, ð‘¥ > 0 3. The total profit (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by ð‘ƒ(ð‘¥) = −ð‘¥^3 + 9ð‘¥^2 + 120𑥠− 400, ð‘¥…
A bag contains three red marbles, one green one, one lavender one, two yellows, and three orange marbles. HINT [See Example 7.] How many sets of four marbles include none of the red ones? A bag contains four red marbles, two green ones, one lavender one, two yellows, and two orange marbles.…
A rifle is shot with an initial velocity of 3,900 ft/sec. If the rifle is fired from a height of 5 ft and aimed at a target with a bullseye at 4 feet high and 1.25 miles away, at what angle of inclination must the rifle be fired to hit the bullseye? Write the answer in degrees, rounded to three…
a) ð‘“(ð‘¥) = ð‘¥^2 + 5ð‘¥. Explain why the Mean Value Theorem applies on [-2, 3]. Find all applicable values of c in (-2, 3). b) Locate the absolute extrema for f(x) = sec x on the interval [−ðœ‹/6 , ðœ‹/3 ]. c) Using the function f(x) = x^3 – 8x and a value of x = 5, find both Δy…
Consider a Leontief economy consisting of two sectors: milk and cheese. Workers can either be assigned as cheesemakers, to produce one pound of cheese each, or milk-makers, to produce one litre of milk each. Every worker must be fed half a pound of cheese and half a litre of milk to stay…
4. Use Newton's method with initial approximation x1 = -1 to find x2, the second approximation to the solution of the following equation. x^3 + x + 5 = 0 x2 = ____________ 5. A stone is dropped from the upper observation deck of a tower, 250 m above the ground. (Assume g = 9.8…
An open top box with height y and a square base with side length x has a volume of 665.5 cm3. Our goal is to find the dimensions x and y that minimize the amount of material used. (1st) Find a formula for the surface area A of the box in terms of only x, the length of one side of the square…
Flight of a golf ball. A golf ball is driven down a horizontal fairway with an initial speed of 55 m/s at an initial angle of 25 degrees (from a tee with negligible height). Neglect all forces except gravity and assume that the ball's trajectory lies in a plane. a.) How far does the ball travel…
The distance s (in m) above the ground for a projectile fired vertically upward with a velocity of 44 m/s as a function of time t (in s) is given by s = 44t - 4.9t^2. Find the answers to these questions. Find t for v = 0. Find v for t = 4. Find v for t = 5. What conclusions can you draw?
A steel ball of mass 3 kg is dropped from rest from the top of a building. If the air resistance is 0.01v and the ball hits the ground after 3.1 seconds, how tall is the building? Answer in four decimal places.
The zoo has 1200 feet of fencing available and wants to build two identically sized rectangular enclosures along one of their buildings. They want to enclose the largest total area possible but aren't sure whether to configure the pens like configuration (A) or (B). They don't need to have…
1. A cube of ice is melting so that its volume is decreasing at a rate of 10 cm3/s. Find the rate at which its surface area is decreasing when its volume is 125 cm3 2. A farmer wants to build 3 pens along a river. The side along the river does not need fencing. If she only has 400m of fencing,…
(b) Using rectangles, each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using the first two and then four rectangles. f(x) = 7/x between x = 5 and x = 9 (c) A silo…
A gun has a muzzle speed 152 m/s. Find two angles of elevation that can be used to hit a target 800m away. Round your answers to one decimal place.
1a) In a rectangle, the length is decreasing at a rate of 0.5 cm/s while the width is increasing at a rate of 0.25 cm/s. At what rate is the area of the rectangle changing when the length is 100 cm and the width is 60 cm? Is the area increasing or decreasing at that time? 1b) Use linear…
(a) Find the total number of bit strings of length 10 that contain exactly seven zeros. (b) Consider the sets A = {a, b, c, d}, B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Compute the total number of functions f: A → B that are injective.
What is the contrapositive of "For all integers n, if n is prime, then Ï•(n)=n−1"? a. There exists an integer n such that if Ï•(n)≠n−1, then n is not prime. b. For all integers n, if Ï•(n)≠n−1, then n is not prime. c. There exists an integer n such that n is prime and…
Find the local extrema of the following function. f(x,y) = 91 + 2xy - 5x2 - 6y2 PART A) Identify the location of any local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There are local maxima located at (Type an ordered pair. Use a…
2. A ball is thrown vertically upward from a height of 4 ft with an initial velocity of 80 ft/s. The distance s (in feet) of the ball from the ground after t seconds is given by s = 4 + 80t - 16t^2. (a) What is the velocity v, in feet per second, of the ball after 2 seconds? v(2) = ft/s (b)…
Reserve Problems Chapter 4 Section 5 Problem 2 A laptop company claims up to 11.1 hours of wireless web usage for its newest laptop battery life. However, reviews on this laptop shows many complaints about low battery life. A survey on battery life reported by customers shows that it follows…
Text 1: Determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x^2 - 9) / (x - 2), [-3, 3] Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in…
a) A 70-lb weight hangs from cables that make the angles of 20° and 67°, respectively, with the horizontal. Find the magnitudes of the forces of tension T1 and T2 in the cables if the resultant force acting on the object is zero. (Round to two decimal places.) b) An 89-lb box is at rest on a…
1. A monopolist sells apples with the demand function q = 24 - p, and the cost of selling q apples is q^2. What value of q maximizes the monopolist's profit? 2. A monopolist can sell oranges for $4 each and the monopolist pays a cost of (1/12)q^3 to sell q oranges. What value of q maximizes the…
(a) Estimate the area under the graph of f, the x-axis, and the lines x = 1 and x = 2 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.) (b) Repeat part (a) using left endpoints.
Texts: (8 points) Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is C(x) = 3000 + 4x + 0.01x^2 + 0.0003x^3. (a) Use C(x) to determine the total cost to produce 120 pairs of jeans and the exact cost to produce just the 121st pair of jeans. Round each…
An incumbent firm (player 1) is either a low-cost type ø1 = L or a high-cost ø1 = H, with equal probability. In period t = 1, the incumbent is a monopolist and sets one of two prices pL or pH, and the profits in this period depend on the incumbent's type and the price it chooses, according to…
The Audubon Society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students $1.10 per serving of pasta. The society estimates that the total cost of producing x servings of pasta at the event will be C(x) = 340 + 0.10x + 0.002x^2…
Suppose the acceleration due to gravity on the moon is -5.5 ft/sec^2, the equation of motion for an object moving in a vertical line near the moon's surface is given by s(t) = -2.75t^2 + v0t + s0, where the subscripts indicate initial conditions. An astronaut on the moon drops a stone from the…
(2) Find the area between the curves y=e^(-0.4x) and y=1.8x+1 from x=0 to x=4 (3) Set up the definite integral required to find the area of the region between the graph of y=17-x^2 and y=2x-18 over the interval -5 ≤ x ≤ 3 (4) Set up the definite integral required to find the area of the…
1) Find the dimensions of the open-topped box with a square base that has a volume of 4 in³ and requires the minimum amount of material to build. (Hint: let x be the length of the base. Express the areas of the base and of four other sides in terms of x. Minimize the sum of these areas.) The…
Part A How much time does it take light to travel from Sirius to The Earth, at a distance of 8.61 light years? Express your answer in years to three significant figures. Part B Light from the Moon takes 1.28 secs to reach the earth. What is the distance to The Moon in kilometers? Express your…
a) If A=(7,-8,-8) and M=(7.5,-5.5,-5) where M is the midpoint of the line segment AB, find the vector in AB. b) A force F of 34N acts on a box in the direction of the vector OP, where P(-1,1,7) and O(0,0,0). Express the force as a vector and find the angle between force F and the xy-plane.
a) Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x+4)/(x-2), [-6, 6] - Yes, the Mean Value Theorem can be applied. - No, f is not continuous on…
A. At a time ð‘¡ seconds after it is thrown up in the air, a tomato is at a height (in meters) of ð‘“(ð‘¡) = -4.9ð‘¡^2 + 35ð‘¡ + 1m. B. What is the average velocity of the tomato during the first 2 seconds? (Include units.) C. Find (exactly) the instantaneous velocity of the tomato at ð‘¡…
Answer the following The Student Society at The Huge State University (ESU) is planning their annual "Come-to" fundraiser. The society will charge students $ 1.75 for pasta service. The society estimates that the total cost of producing x pasta services in the event will be C (x) = 310 + 0.07 x…
Answer the following The Student Society at The Huge State University (ESU) is planning their annual "Come-to" fundraiser. The society will charge students $ 1.75 for pasta service. The society estimates that the total cost of producing x pasta services in the event will be C (x) = 310 + 0.07 x…
A dune buggy is on the desert at a point A located 40 km from a point B, which lies on a long straight road. The driver can travel at 45 km/h on the desert and 75 km/h on the road. The driver will win a price if she arrives at the finish line at point D, 50 km from B, in 84 min or less. What…
A particle is moving with the given data. Find the position of the particle. a(t)=sin t+cos t, s(0)=0, v(0)=6
Power Series
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Suppose a baseball is thrown vertically upward from the surface of a planet. Its height (in meters) after t seconds is s(t) = (30)t - (0.357)t^2. Use calculus to find: How many seconds will it take for the ball to reach the maximum height?
3. A cake shop needs to make boxes in a cylinder shape. Each box has to have a volume of 10π m3. The materials for the top and bottom of the box costs 6$/m2 and materials for the side of the box costs 4$/m2. Find the radius of the box that will minimize the building costs.
2-Pure water, 4 times its volume, is added to the 30% alcohol solution. What is the new alcohol percentage? 4-There are 60g of A, 30g of B, and 20g of substance C in the 110g X blend. How many grams of X mix can be made with 360g of A, 400g of B, and 500g of substance C? 6-6 apprentices do 3…
a) Find the local maximum and minimum values and saddle points of the function f(x,y) = x^2y^2 - 8x + y/xy. Graph the factors (roughly) with a domain of view point that reveal all the important aspects of the function. f_x = y - x^(-2) = 0 f_y = x + 8y^(-2) = 0 b) A cardboard box without a…
According to a study, the probability that a randomly selected teenager studied at least once during the week was only 0.54. Let X be the number of teenagers who studied at least once during the week. What is the probability that only 4 students from your study group of ten have studied this…
stone height: 112 tokens <thought
PROMPT: (Problem, PART ONE): A motorboat travels 80 miles in 2 hours going upstream. It travels 112 moles going downstream in the same amount of time. What is the rate of the boat in still water and what is the rate of the current? RATE IN THE BOAT IN STILL WATER: ____ mi/h RATE OF THE CURRENT:…
a) A printed page is to have 60 cm² of printed material with 5 cm margins on the sides and 3 cm margins on top and bottom. What is the minimum total area of the page? b) An open-topped storage box with a square base is to have a capacity of 5 m³. Sides cost $1.60/m² while material for the…
1. Find ∂f/∂x where f(x, y) = sin(x/y). (a) (x/y) cos(x/y). (b) (-x/y)^2 cos(x/y). (c) ((y - x)/y)^2 cos(x/y). (d) (1/y) cos(x/y). (e) None of the other choices. 2. Find ∂f/∂y where f(x, y) = sin(x/y). (a) (x/y) cos(x/y). (b) (-x/y^2) cos(x/y). (c) ((y - x)/y^2) cos(x/y). (d) (1/y)…
A cable runs along the wall from C to P at a cost of $5 per meter, and straight from P to M at a cost of $13 per meter. Let x be the distance from C to P. If M is 24 meters from the nearest point A on the wall where P lies, and A is 53 meters from C, find x such that the cost of installing the…
Consider the following statement: if for every p ∈ Q, there exists q ∈ Q such that |p - q| < 1, then 2^10 ≤ 10^3. (a: 4 points) Write down the negation of the boldface statement. (The boldface statement contains English words; so should your answer.) (b: 4 points) Write down the converse…
Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12 gives a remainder of 7, what is the remainder when 5b is divided by 12? Fill in the blanks to show that the same answer will be obtained no matter what integer is used for b at the…
Suppose the distance s (in feet) covered by a car moving along a straight road after t seconds is given by the function s = f(t) = 3t^2 + 13t. Calculate the instantaneous velocity of the car when t = 35 s. Group of answer choices: 4130 223 350 445 2. If a rock is thrown upward on the planet…
Find the x-values where points of inflection occur. State the x-intervals where the function is concave up and the x-intervals where it is concave down. f(x) = -x^3 + 9x^2 - 55 What are the x-values of the inflection point(s)? A. The x-values of the inflection point(s) are x = (Use a comma…
Find the center of mass of the given system of point masses lying on the x-axis. m1 = 6, m2 = 3, m3 = 5 x1 = -5, x2 = 0, x3 = 3
A gun with a muzzle velocity of 600 m/s is fired at an angle of 12° above the horizontal. Find the horizontal and vertical components of the velocity.
A study of a country's colleges and universities resulted in the demand equation q = 20,000 - 2p, where q is the enrollment at a public college or university and p is the average annual tuition (plus fees) it charges. Officials at Enormous State University have developed a policy whereby the…
A rock is dropped off of a tall building. Its height (in feet) after t seconds is given by s(t) = -16t^2 + 1000. a. What is its velocity 1 second after being dropped? Include units. Show work. b. What is its acceleration 1 second after being dropped? Include units. Show work.
Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f(x) = 2x^4 - 2x^3 + 5 Determine the intervals on which the given function is concave up or concave down. Select the correct choice below and fill in the answer box(es) to…
Victor is on a mountain whose height at the location (x, y) is given by z = x(6 + sin y). If Victor is at (1, π/2), in what direction should he take to get down the mountain as fast as possible?
Complete the following steps for the given integral and the given value of n. a. Graph the integrand on the interval of integration. b. Calculate Δx and the grid points x0, x1,...,xn assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d.…
The number of Americans age 65 or older can be modeled as n(x) = -0.00082x^3 + 0.059x^2 + 0.183x + 34.42, where output is measured in million people and x is the number of years since 2000, projections through 2050.†(a) What is the projected number of Americans 65 years of age and older in…
In the diagram shown, a puck A of mass 0,024 kg is moving to the right along the x-axis with a speed of 6.08 m/s. It collides with puck B of mass 0.048 kg initially at rest. The collision is not head-on. After the collision, the two pucks fly apart in the directions shown in the…
1. Find the length of the curve y = , 0 ≤ x ≤ 1 Length of curve = 2. The mean value of a function is defined on the interval [a , b] as Find the mean value of f(x) = on the interval [0,].
Suppose you are on a ferris wheel with a diameter of 20 m and making one turn per minute. How fast do you rise or fall when you're 6 m horizontally from the vertical line that runs through the center of the wheel?
A car traveled 50 miles due south then traveled 90 miles 35 degrees south of west. Solve for the resultant displacement and its direction.
In your company, the production costs in SRs per week of producing x units is given by C(x) = 3x + x^2 + 6x^3 and the demand function for the units is given by p(x) = x - x^2. What is the marginal cost function, marginal revenue function, and marginal profit function?
Approximate the area under f(x) = x^3 - 8x^2 + 15x + 10 from x = 1 to x = 5 using (a) 6 rectangles and starting with the right endpoint (R6) (b) 3 rectangles and starting with the left endpoint (L3) (c) Compare both of your approximations with the exact area under the curve and determine if…
Find the critical points of the function f(x) = 5x^2 - 10x. Apply Rolle's theorem for the function f(x) = x^2 + 5x over the interval [-5,0]. Justify your answer. Verify the Mean Value Theorem for the function f(x) = 2x^3 + x + 5 over the interval [0,1]. A ball is dropped from a certain height.…
A model rocket is vertically launched and its position is modeled by s(t) = -5t^2 + 10(2)(5)t + 3, where s is measured in meters and t is in seconds. Determine the equations of velocity and acceleration. b) Determine its initial position, initial velocity, and initial acceleration. c) Determine…
A particle moves in a straight line with the given velocity ð‘£(ð‘¡) = 3ð‘¡Â² - 30ð‘¡ + 27 (in m/s). Find the displacement and distance traveled over the time interval [0,10]. (Give your answers as whole or exact numbers.) Displacement: Total distance traveled:
Suppose a small (radius zero) ball is rolling along the curve 𑦠= 18/x³ + 1 with a constant speed of 3 ð‘š/ð‘ . What is the velocity vector of the ball when it passes through the point (2,2)?
A linear cost function is C(x) = 7x + 700 (Assume C is measured in dollars.) (a) What are the slope and the C-intercept? slope: 7 C-intercept: 700 (b) What is the marginal cost MC? MC = 7 What does the marginal cost mean? a) Each additional unit produced reduces the cost by this much (in…
Suppose that a survey of 100 advertisers in U.S. News, These Times, and World found the following: - 15 advertised in all three - 28 advertised in These Times and U.S. News - 31 advertised in World and U.S. News - 29 advertised in World and These Times - 59 advertised in These Times - 55…
Consider the curve c(t) = <t, sin(3t), cos(3t)>. a. Compute the velocity and acceleration. b. Find the length of the curve over the interval [0, π]. c. Find the curvature κ.
An object is propelled vertically upward from the ground with an initial velocity of 43.4 m/s. The signed distance s (in meters) of the object from the ground after t seconds is given by the position function s = s(t) = -4.9t^2 + 43.4t. (a) What is the velocity, in meters per second, of the…
A kite 50 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 150 ft of string has been let out? Answer (in radians per second):
Baggzz, Inc. sells custom backpacks to the students of a community college in California's San Joaquin Valley. It determines that in order to sell x backpacks, the price per backpack must be p = 150 - 0.75x. It also determines that the total cost of producing x backpacks is given by C(x) = 3000…
A point has horizontal and vertical displacements (in cm) of x = t3 - t2 + 5t and y = 24 - t3 + t. Find the magnitude and direction of the resultant velocity at t = 3.50 s.
a kite is flying 40 ft above the ground at the end of a string. the girl flying the kite lets out the string at a rate of 1 ft per second. If the kite remains 40 ft above the ground How fast the horizontal distance from the girl to the point below it increasing when 50 ft string is out
The cost of producing x teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula C(x) = 100 + 39x - 0.06x^2. (a) Find the marginal cost function C'(x). C'(x) = Use it to determine how fast the cost is going up (in $) at a production…
A projectile if fired with an initial speed at 100m/s and an angle of elevation of 45 degrees. A) find the position vector function, r(t), for the projectile. B) Find the range of the projectile. C) Find the maximum height reached.
A piece of wire 13 m long is cut into two pieces. One piece is bent into the shape of a circle of radius ð‘Ÿ and the other is bent into a square of side ð‘ . How should the wire be cut so that the total area enclosed is: a) a maximum? ð‘Ÿ= and ð‘ = . b) a minimum? ð‘Ÿ= and ð‘ =.
1a) A power house, P, is on one bank of a straight river 200 m wide, and a factory, F, is on the other bank 400 m downstream from P. The cable has to be taken across the river under water at a cost of $12.00/m. On land the cost is $6.00/m. What path should be chosen so the cost is…
Consider the following cost and revenue functions for a city pair route that an airline is the single seller for: C(q) = 0.001q^3 - 1.8q^2 + 1180q + 134000 R(q) = -0.859q^2 + 1374.4q Required: A) Develop a profit function B) At what value of q are profits at a maximum? C) What is the revenue…
The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The marketing department has determined that the demand for these speakers is represented by the following function, where p denotes the speaker's unit price (in dollars) and x denotes the quantity…
1a.) Suppose that an object moves along an s-axis so tha its location is given by s(t)= t^2 + 5t at t. (Here s is in meters and t in seconds) Find the average velocity of the objects in meters per second over the time interval t=2 to t=8 seconds. 1b.) Find the instantaneous velocity of the…
Find a cubic polynomial f(x) = ax^3 + bx^2 + cx + d for the landing of a fighter jet. The function f(x) will be the height of the plane from the ground, and x the ground distance of the plane away from the airport. Everything is measured in miles. The fighter jet is cruising at 35,000 feet and…
A rock is dropped from the edge of a cliff that is 320 meters above the water. What is its velocity at impact? (Hint: Use the height function h(t) = 5t^2 + v0t + s0 for free-falling objects.)
You are stationary on the ground and are watching a bird fly horizontally towards you at a rate of -6 m/s. The bird is located 60 m above your head. How fast does the angle of elevation of your head change when the horizontal distance between you and the bird is 160 m? (Leave your answer as an…
a) State the Mean Value Theorem. b) Sketch a graph that illustrates the conclusion of the Mean Value Theorem.
A projectile is shot upward from the surface of Earth with an initial velocity of 111 meters per second. Use the position function below for free-falling objects. What is its velocity after 2 seconds? After 14 seconds? (Round your answers to one decimal place.) s(t) = -4.9t^2 + v0t + s0 s =…
The government is trying to get more teenagers in juvenile detention to go to school while they are in detention. The management at the facility wants to test if cash payments will increase school attendance. They randomly split the children (PP) into a treatment and control group (PP) where…
a) A manufacturer estimates that the hourly cost of producing x units of a product on an assembly line is C(x) = 0.2x^3 - 8x^2 + 146x + 300 dollars. b) What is the change in cost when production changes from 10 units to 50 units? Include units. c) What is the average rate of change of cost when…
5. Find the area bounded by the graphs of the indicated equations over the given interval. y = e^x; y = -1/x; 1 ≤ x ≤ 2 The area is ? square units. 6. Find the consumers' surplus and the producers' surplus at the equilibrium price level for the given price-demand and price-supply…
Find the intervals on which the function is continuous. 6) y = 4√(10x - 10) A) (-∞, 1) B) (1, ∞) C) (1, ∞) D) (-∞, 1) Find dy/dx. 7) s = t^3 - csc(t) + 17 A) 3t^2 - csc(t) cot(t) B) t^2 - cot^2(t) + 17 C) 3t^2 + cot^2(t) D) 3t^2 + csc(t) cot(t) Find dy/dx by implicit…
A rock is thrown from a Martian cliff which has a height of 288 feet. The initial velocity of the rock is 9 feet per second. Its speed in feet per second at time t seconds after it is thrown is given by the equation: s(t) = 3t + 9 How far is the rock from the ground 3 seconds after it is…
Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = 5x^2 - 4x + 1, [0, 2] Yes, it does not matter if f is continuous or differentiable, every function satisfies the Mean Value Theorem. Yes, f is continuous on [0, 2] and differentiable on (0, 2) since…
A baseball player at second base throws a ball 90 feet to a player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 50 miles per hour and at an angle of 15 degrees above the horizontal. At what height does the player at first base catch the ball?
Draw a Markov chain with at least 4 states, and then write the matching transition matrix. Be sure to explain what the situation is that your Markov chain is modeling. What do each of the states mean? Why do the probabilities make sense?
Consider the function ln x. a. Normalize it so that it is a PDF on [2, 5]. b. Compute the expected value of the PDF on that interval. c. Compute the variance of the PDF on that same interval.
20.) At a price of $60 for a 4-hour city tour, a bus tour company attracts 120 customers. For every $2 decrease in price, an additional four customers are attracted. The cost, in dollars, for accommodating ð‘¥ customers on the tour is: ð¶(ð‘¥) = 0.1ð‘¥^2 + 12ð‘¥ + 1500 a. Find the demand…
A lamina has the shape of the planar region bounded by y = x^2, y = 0, x = 0, and x = 1. The density at each point of the lamina is given by p(x, y) = x^2 + 2y. Find the mass m of the lamina and the x-coordinate x* of the center of mass of the lamina.
Let f be the function given by f (x) = x3 – 6x2 . What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [0, 6]. ( A ) 0 only ( B ) 4 only ( C ) 0 and 4 (D ) 4 and 6 ( E ) None of these
Approximate the area under the curve for the function f(x) = 9x^3 from x = 0 to x = 3 by summing the areas of 'thin' rectangles. a) Use n = 3 rectangles (of the same width). a1) Find a left-endpoint sum by using the left endpoint for the height of each rectangle. a2) Find a right-endpoint sum…
An object is launched at an angle θ = π/4 from the horizontal with an initial speed of 80 m/s. Determine the time of flight and the horizontal range. The object is launched from the ground, and air resistance is ignored.
A gun is fired at an angle of 30 degrees above horizontal. What is the muzzle speed (initial speed) of the bullet if the maximum height attained by the bullet is 500 meters?
A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure. Let r represent the radius of the circle and h the height of the rectangle. To ensure that the window admits enough light, the area of the window must be 90 square feet. (a) Write a formula…
As shown below, an object weighing 300 N is lying on an inclined plane. 45° (a) Write the force of gravity as a vector in 2-space. (b) Use projections to find the vector component of the force of gravity acting parallel to the ramp. Do not rotate the axes! (c) If the object moves 40 m down the…
Given r = f(theta) = 1 + sin(theta). (15 Points) (a) Graph the curve. (b) Find the area bounded by the curve. (c) Find the points on the curve where the tangent line is horizontal.
Professor Nimrod throws a piece of chalk, at an angle of 39° above the horizon, with initial velocity of 28 feet per second. Given that the chalk was released from a point 6 feet above the ground, find the parametric equations that model the trajectory of the chalk. Also, there is a wall 10…
1. a. Each side of a square is increasing at a rate of 2 cm/s. At what rate (in cm2/s) is the area of the square increasing when the area of the square is 16 cm2? b. The length of a rectangle is increasing at a rate of 7 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 9…
18. what is the maximum area that a rectangle inscribed in a semicircle of radius 13_ / 2 can have? a.13 square centimeters b. 169 square centimeters c. 26 square centimeters d. 338 square centimeters e. 676 square centimeters
(10 points) Suppose that X and Y have the following joint probability distribution: f(x, y) x 0 1 2 0 y 1 2 3/28 9/28 3/28 3/14 3/14 0 1/28 0 0 Evaluate the marginal…
Suppose R(x) dollars gives the revenue from the sale of Clemson hats, when x hats are produced and sold. a). Write a sentence of interpretation explaining that the marginal revenue is 5.28 dollars when 40 hats are sold. (don't use the word "marginal") "When 40 Clemson hats are produced and…
Alex needs to buy building supplies for his new patio. He needs 20 bags of cement, 50 cubic feet of sand, and 100 red bricks. There are two building supply stores in town, Rocko's and Big Mike's. The prices for each of the items are shown in the table. Cement: - Rocko's: $4.00 per bag - Big…
The marginal revenue function of the company is G'(x) = -0.4x + 280, in dollars per unit, when x units are sold. 1- Determine the G(x) revenue function of the company. 2- Find the revenue attained by the company when the first 60 units are sold. 3- Find the additional revenue realized by the…
Suppose a stone is thrown vertically upward from the edge of a cliff with an initial velocity of 96 ft/s from a height of 10 ft above the ground. The height s (in ft) of the stone above the ground t seconds after it is thrown is s = -16t^2 + 96t + 10. a) Determine the velocity v of the stone…
a ball weighing 150 newtons lies on an inclined ramp of height 50 m and base 110 m. if the ball slides 15 m down the ramp, calculate the work done in Nm by the force of gravity acting on the ball.
5. Person A earns $40 per hour and Person B earns $42 per hour. ln 40 = 3.689 and ln 42 = 3.738 a. Find the exact percentage by which Person B's hourly salary exceeds Person A's. b. Using logarithms, find the approximate percentage by which Person B's salary exceeds Person A's. 6. Fill in the…
USE CALCULUS, not just physics trig!!! Scientists discover a new planet, Zed. On Zed, the acceleration due to gravity is 32 m/s^2. A bullet is shot from the ground on Zed, at an angle of pi/6 from the ground. The bullet is shot at a velocity of 800 m/s. USING CALCULUS a) derive an equation for…
(6 points) The table below gives the population (in thousands of residents) of a coastal community for the period from 1990 to 1999. Figures shown are for January 1 in each year. Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Population 618 841 1322 1550 1621 1487 957 801 768 738 (a).…
A rectangle is constructed with its base on the diameter of a semicircle with radius 25, and with its two other vertices on the semicircle. Let x be the length (base) of the rectangle that lies on the diameter of the semicircle. Find the dimensions of the rectangle with maximum area. Draw the…
The demand curve for a product is given by q = 144,000 - 600p and the supply curve is given by q = 1000p for 0 ≤ q ≤ 144,000, where price is in dollars. (a) At a price of $100, what quantity are consumers willing to buy and what quantity are the producers willing to supply? (b) Will the…
A stone is thrown vertically upward from the top of a building 64 ft above the ground with an initial velocity of 48 ft/sec. Ignore air resistance. Determine how long it takes for the stone to strike the ground and with what velocity it will strike the ground.
1. Use algorithm of curve sketching to sketch the graph of f(x) = x4 + 4x3. 2. A mass of 5 kg is being suspended by two strings, 24cm and 32cm, from two points that are 40cm apart at the same level. Determine the tension in each of the strings.
A farmer plans to enclose a rectangular pasture adjacent to a river (see figure). The pasture must contain 5,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. What dimensions will require the least amount of fencing? x=_______m y=_______m
Find the sum of the maximum and minimum values of the function f(x,y,z) = 6x + 12y + 18z on the ellipse that is formed by the intersection of the cylinder x^2 + y^2 = 36 and the plane y + z = 6.
Properties of Probability, Permutations, Combinations Where applicable, show the permutations/combinations you set up to get the answer. The superintendent of a certain school district has estimated the probabilities associated with the SAT verbal scores of students from that district. The…
Problem three: Let f(0) = 0 = g(1), f(1) = 2 = g(0), and f'(0) = 1, g'(1) = 2. f(t) - g(t + 1) (a) Evaluate: lim (b) Find the derivative of (fog)(r) at r = 1. Problem four: Find dr for see (ry) = r² + y? - 13. Problem five: Consider the function f(x) = cos(2r) + 2x. re [0, T]. (a) Find…
Answer the following questions for the function f(x)=−2x3−3x2+72x−10. a. Find formulas for f′(x) and f′′(x). f′(x)= ____ f′′(x)= ____ Enter f(x), f′(x), and f′′(x) into your grapher to examine the table. b. The formula for the first derivative f′(x) can be factored. Set f′(x)=0 to find the two…
A rock is dropped from a height of 89.5 m and falls toward Earth in a straight line. In t seconds the rock falls 4.9t2 meters. What is the velocity v of the rock when it hits the ground? (Round your answer to two decimal places.)
A ball is thrown at an angle of 60 ∘ to the ground. If the ball reaches the maximum height in 2 seconds, what was the initial speed of of the ball? How far does the ball fly?
There is a jet which is flying in the north direction at a speed of 800 km/h. It then faces a wind blowing in the southeast direction. The wind has a speed of 90 km/h. Calculate the resultant velocity of the airplane. Also, draw a diagram to support your explanation.
f(x) = -x^3 + 3x^2 - 4 1.b. What is the y-intercept of f(x)? 1.c. What are the x-intercepts of f(x)? Hint: (x + 1) is a factor, use synthetic division to find the other factors. 1.d. Find the relative extrema of f(x). State your answers as a coordinate point or "none". Relative Minimum:…
1. A contractor has installed a silt fence around an area that is semi-circular and level to prevent soil from the construction site entering nearby streams. The diameter of the semi-circle is 1,650 feet. How many linear feet of fence does the contractor need to install to enclose the area? 2.…
2) f(x,y) = 5 over the rectangle R = {(x,y): 2 ≤ x ≤ 3, 2 ≤ y ≤ 4} Value of the integral of the constant function is? a) 20 b) 25 c) 50 d) 10 3) f(x,y) = x^2 + y^2 over the unit circle. How to represent the integral form?
Suppose f is continuous on a closed interval [a, b]. If u is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = u. A. Let g(x) = 1/x. Choose an interval where the Intermediate Value Theorem does not apply and explain why. Then, choose a u value for…
8. The height, y, of a ball that is tossed straight up is given by the following formula: -16t^2 + 35t + 15, where the height is measured in feet and t is the time since release measured in seconds. (Negative represents the downward direction.) Include units on all answers. A. At what height…
1. The cost function C(x) = 15,000 + 30x + 1,000/x gives the cost in dollars to produce x units of a commodity. a. Find the marginal cost when the production level is x = 100 units. When they produce 100 units, the marginal costs is ____________ In other words, the approximate cost to produce…
3. Consider the function below. f(x)= (x^2)/(x^2 - 16) (a) Find the vertical asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) x = ? Find the horizontal asymptote(s). (Enter your answers as a comma-separated list. If an answer does not exist,…
Find the value of k such that the function f(x) = kx^3 is a probability density function on the interval [1,∞). Group of answer choices: a. 3 b. 4 c. 2 d. 1 2. The function f(x) = 1/(4√x) is a probability density function on the interval [1,9]. Find the following (round to 4 decimal places, if…
1. Sketch the graph of a function f that is continuous except that it has a removable discontinuity at x = 3 and a jump discontinuity at x = 5. Explain your answer. 2. If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H =…
A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial…
11. Determine the absolute minimum of f(x) = -5x^3 - 12x^2 + 39x + 6 on the interval [-1,2] a) 4 b) -28 c) -40 d) -48 15. Find the area on the x-axis, under the curve of f(x) = 3sec^2(2x/5) - 7 between x = 1 and x = 2 Select one: a. 10.57 b. 10.96 c. 11.37 d. 11.55 17. Estimate the area under…
Two forces of magnitude 80N and 45N act on an object, with an angle between the forces of 100 degrees . What is the resultant force?
A company plans to enclose three parallel rectangular areas for sorting returned goods. The three areas are within one large rectangular area and 992 yd of fencing is available. What is the largest total area that can be enclosed? A rectangle has 2 horizontal sides and 2 vertical sides. Two…
You are shooting a tennis ball, straight up, off the top of a 36 foot high platform using a tennis ball cannon. The cannon can shoot the ball at 36 feet per second. Answer the following questions: (a) How fast is the ball traveling at highest point? (b) At what time is the ball at its peak…
a) An academician wants to see significant mean differences in the performance level of private and public banks in Kabul, Afghanistan. You have been hired as a consultant to suggest a specific quantitative test. Which test would you propose and why? b) In another situation, a manager wants to…
Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 2 (a) Find dy/dt, given x = 2 and dx/dt = 14. dy/dt = (b) Find dx/dt, given x = 1 and dy/dt = –7. dx/dt =
1. Locate the centroid of the area bounded by the x-axis and y = -5 + 3x^2. a. (0, -9/5) b. (0, -5/2) c. (0, -3) d. (0, -2) 2. Find the 10th term in the expansion of (b - 3c)^12. a. 1,299,078b^2c^10 b. -1,299,078b^3c^9 c.…
I conducted a survey among my 58 students in my evening statistics online class to see who eats spaghetti at least twice a week. Based on the frequency, I find that 40% of them eats spaghetti at least twice a week. Therefore, I make a same conclusion for all students from my university. Assume…
Suppose that: f(x)= (4)/(x^2-36) (A) Find all critical values of f, compute their average, and enter it below. Note: If there are no critical values, enter -1000. (B) Use interval notation to indicate where f(x) is increasing. (C) Use interval notation to indicate where f(x) is decreasing. (D)…
Text: 3 Outliers In many statistical analysis programs, data values that are considerably outside the range of the majority of the values are simply dropped from consideration. Using this information, write a program named outliers.py that accepts a number of floating-point values from the user…
A cannonball is launched at 32 degrees with a velocity of 130 m/s. Write vector equations for the trajectory, the velocity, and the acceleration, and determine the flight time, maximum height, and range for a ground-to-ground shot.
2. Consider the function g(x) = 1 - 1/x + 1/x^2. (a) Find the domain and x, y-intercepts. (b) Find the asymptotes (if they exist). (c) Find the intervals of increasing/decreasing and local extrema. (d) Find the intervals of concavity and inflection points. (e) Sketch the curve. (Note: You may…
2. Find the absolute extrema, if they exist, as well as where they occur for ð‘“(ð‘¥) = 2ð‘¥ + 8ð‘¥^2 + 1, ð‘¥ > 0 3. The total profit (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by ð‘ƒ(ð‘¥) = −ð‘¥^3 + 9ð‘¥^2 + 120𑥠− 400, ð‘¥…
A rifle is shot with an initial velocity of 3,900 ft/sec. If the rifle is fired from a height of 5 ft and aimed at a target with a bullseye at 4 feet high and 1.25 miles away, at what angle of inclination must the rifle be fired to hit the bullseye? Write the answer in degrees, rounded to three…
a) ð‘“(ð‘¥) = ð‘¥^2 + 5ð‘¥. Explain why the Mean Value Theorem applies on [-2, 3]. Find all applicable values of c in (-2, 3). b) Locate the absolute extrema for f(x) = sec x on the interval [−ðœ‹/6 , ðœ‹/3 ]. c) Using the function f(x) = x^3 – 8x and a value of x = 5, find both Δy…
Consider a Leontief economy consisting of two sectors: milk and cheese. Workers can either be assigned as cheesemakers, to produce one pound of cheese each, or milk-makers, to produce one litre of milk each. Every worker must be fed half a pound of cheese and half a litre of milk to stay…
An open top box with height y and a square base with side length x has a volume of 665.5 cm3. Our goal is to find the dimensions x and y that minimize the amount of material used. (1st) Find a formula for the surface area A of the box in terms of only x, the length of one side of the square…
Flight of a golf ball. A golf ball is driven down a horizontal fairway with an initial speed of 55 m/s at an initial angle of 25 degrees (from a tee with negligible height). Neglect all forces except gravity and assume that the ball's trajectory lies in a plane. a.) How far does the ball travel…
The distance s (in m) above the ground for a projectile fired vertically upward with a velocity of 44 m/s as a function of time t (in s) is given by s = 44t - 4.9t^2. Find the answers to these questions. Find t for v = 0. Find v for t = 4. Find v for t = 5. What conclusions can you draw?
The zoo has 1200 feet of fencing available and wants to build two identically sized rectangular enclosures along one of their buildings. They want to enclose the largest total area possible but aren't sure whether to configure the pens like configuration (A) or (B). They don't need to have…
1. A cube of ice is melting so that its volume is decreasing at a rate of 10 cm3/s. Find the rate at which its surface area is decreasing when its volume is 125 cm3 2. A farmer wants to build 3 pens along a river. The side along the river does not need fencing. If she only has 400m of fencing,…
(b) Using rectangles, each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using the first two and then four rectangles. f(x) = 7/x between x = 5 and x = 9 (c) A silo…
A gun has a muzzle speed 152 m/s. Find two angles of elevation that can be used to hit a target 800m away. Round your answers to one decimal place.
1a) In a rectangle, the length is decreasing at a rate of 0.5 cm/s while the width is increasing at a rate of 0.25 cm/s. At what rate is the area of the rectangle changing when the length is 100 cm and the width is 60 cm? Is the area increasing or decreasing at that time? 1b) Use linear…
(a) Find the total number of bit strings of length 10 that contain exactly seven zeros. (b) Consider the sets A = {a, b, c, d}, B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Compute the total number of functions f: A → B that are injective.
What is the contrapositive of "For all integers n, if n is prime, then Ï•(n)=n−1"? a. There exists an integer n such that if Ï•(n)≠n−1, then n is not prime. b. For all integers n, if Ï•(n)≠n−1, then n is not prime. c. There exists an integer n such that n is prime and…
Find the local extrema of the following function. f(x,y) = 91 + 2xy - 5x2 - 6y2 PART A) Identify the location of any local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There are local maxima located at (Type an ordered pair. Use a…
2. A ball is thrown vertically upward from a height of 4 ft with an initial velocity of 80 ft/s. The distance s (in feet) of the ball from the ground after t seconds is given by s = 4 + 80t - 16t^2. (a) What is the velocity v, in feet per second, of the ball after 2 seconds? v(2) = ft/s (b)…
a) A 70-lb weight hangs from cables that make the angles of 20° and 67°, respectively, with the horizontal. Find the magnitudes of the forces of tension T1 and T2 in the cables if the resultant force acting on the object is zero. (Round to two decimal places.) b) An 89-lb box is at rest on a…
1. A monopolist sells apples with the demand function q = 24 - p, and the cost of selling q apples is q^2. What value of q maximizes the monopolist's profit? 2. A monopolist can sell oranges for $4 each and the monopolist pays a cost of (1/12)q^3 to sell q oranges. What value of q maximizes the…
(a) Estimate the area under the graph of f, the x-axis, and the lines x = 1 and x = 2 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.) (b) Repeat part (a) using left endpoints.
Texts: (8 points) Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is C(x) = 3000 + 4x + 0.01x^2 + 0.0003x^3. (a) Use C(x) to determine the total cost to produce 120 pairs of jeans and the exact cost to produce just the 121st pair of jeans. Round each…
The Audubon Society at Enormous State University (ESU) is planning its annual fund-raising "Eatathon." The society will charge students $1.10 per serving of pasta. The society estimates that the total cost of producing x servings of pasta at the event will be C(x) = 340 + 0.10x + 0.002x^2…
Suppose the acceleration due to gravity on the moon is -5.5 ft/sec^2, the equation of motion for an object moving in a vertical line near the moon's surface is given by s(t) = -2.75t^2 + v0t + s0, where the subscripts indicate initial conditions. An astronaut on the moon drops a stone from the…
(2) Find the area between the curves y=e^(-0.4x) and y=1.8x+1 from x=0 to x=4 (3) Set up the definite integral required to find the area of the region between the graph of y=17-x^2 and y=2x-18 over the interval -5 ≤ x ≤ 3 (4) Set up the definite integral required to find the area of the…
1) Find the dimensions of the open-topped box with a square base that has a volume of 4 in³ and requires the minimum amount of material to build. (Hint: let x be the length of the base. Express the areas of the base and of four other sides in terms of x. Minimize the sum of these areas.) The…
Part A How much time does it take light to travel from Sirius to The Earth, at a distance of 8.61 light years? Express your answer in years to three significant figures. Part B Light from the Moon takes 1.28 secs to reach the earth. What is the distance to The Moon in kilometers? Express your…
a) If A=(7,-8,-8) and M=(7.5,-5.5,-5) where M is the midpoint of the line segment AB, find the vector in AB. b) A force F of 34N acts on a box in the direction of the vector OP, where P(-1,1,7) and O(0,0,0). Express the force as a vector and find the angle between force F and the xy-plane.
a) Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = (x+4)/(x-2), [-6, 6] - Yes, the Mean Value Theorem can be applied. - No, f is not continuous on…
A. At a time ð‘¡ seconds after it is thrown up in the air, a tomato is at a height (in meters) of ð‘“(ð‘¡) = -4.9ð‘¡^2 + 35ð‘¡ + 1m. B. What is the average velocity of the tomato during the first 2 seconds? (Include units.) C. Find (exactly) the instantaneous velocity of the tomato at ð‘¡…
Answer the following The Student Society at The Huge State University (ESU) is planning their annual "Come-to" fundraiser. The society will charge students $ 1.75 for pasta service. The society estimates that the total cost of producing x pasta services in the event will be C (x) = 310 + 0.07 x…
Answer the following The Student Society at The Huge State University (ESU) is planning their annual "Come-to" fundraiser. The society will charge students $ 1.75 for pasta service. The society estimates that the total cost of producing x pasta services in the event will be C (x) = 310 + 0.07 x…
A dune buggy is on the desert at a point A located 40 km from a point B, which lies on a long straight road. The driver can travel at 45 km/h on the desert and 75 km/h on the road. The driver will win a price if she arrives at the finish line at point D, 50 km from B, in 84 min or less. What…
A particle is moving with the given data. Find the position of the particle. a(t)=sin t+cos t, s(0)=0, v(0)=6
Power Series
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Use ratio test to check weather it converges
show,explain
Show, explain, use, find
Explain, suppose, find
what is the answer?
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