Calculus for Scientists and Engineers: Early Transcendental

William Briggs, Lyle Cochran, Bernard Gillett

Chapter 6

Applications of Integration - all with Video Answers

Educators

Section 1

Velocity and Net Change

Problem 1

Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line.

Bob Gorbold

Bob Gorbold

Numerade Educator

Problem 2

Suppose the velocity of an object moving along a line is positive. Are position, displacement, and distance traveled equal? Explain.

Bob Gorbold

Numerade Educator

Problem 3

Given the velocity function $v$ of an object moving along a line, explain how definite integrals can be used to find the displacement of the object.

Bob Gorbold

Numerade Educator

Problem 4

Explain how to use definite integrals to find the net change in a quantity, given the rate of change of that quantity.

Bob Gorbold

Numerade Educator

Problem 5

Given the rate of change of a quantity $Q$ and its initial value $Q(0)$ explain how to find the value of $Q$ at a future time $t \geq 0.$

Bob Gorbold

Numerade Educator

Problem 6

What is the result of integrating a population growth rate between two times $t=a$ and $t=b,$ where $b>a ?$

Bob Gorbold

Numerade Educator

Problem 7

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=6-2 t ; 0 \leq t \leq 6$$

Bob Gorbold

Numerade Educator

Problem 8

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=10 \sin 2 t ; 0 \leq t \leq 2 \pi$$

Bob Gorbold

Numerade Educator

Problem 9

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=t^{2}-6 t+8 ; 0 \leq t \leq 5$$

Bob Gorbold

Numerade Educator

Problem 10

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=-t^{2}+5 t-4 ; 0 \leq t \leq 5$$

Bob Gorbold

Numerade Educator

Problem 11

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=t^{3}-5 t^{2}+6 t ; 0 \leq t \leq 5$$

Bob Gorbold

Numerade Educator

Problem 12

Displacement from velocity Assume t is time measured in seconds and velocities have units of $m / s$.a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval.
$$v(t)=50 e^{-2 t} ; 0 \leq t \leq 4$$

Bob Gorbold

Numerade Educator

Problem 13

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=\sin t \text { on }[0,2 \pi] ; s(0)=1$$

Kevin Luu

Numerade Educator

Problem 14

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=-t^{3}+3 t^{2}-2 t \text { on }[0,3] ; s(0)=4$$

Kevin Luu

Numerade Educator

Problem 15

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=6-2 t \text { on }[0,5] ; s(0)=0$$

Kevin Luu

Numerade Educator

Problem 16

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=3 \sin \pi t \text { on }[0,4] ; s(0)=1$$

Kevin Luu

Numerade Educator

Problem 17

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=9-t^{2} \text { on }[0,4] ; s(0)=-2$$

Kevin Luu

Numerade Educator

Problem 18

Position from velocity Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for $t \geq 0,$ using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval.
$$v(t)=1 /(t+1) \text { on }[0,8] ; s(0)=-4$$

Kevin Luu

Numerade Educator

Problem 19

Oscillating motion A mass hanging from a spring is set in motion and its ensuing velocity is given by $v(t)=2 \pi \cos \pi t,$ for $t \geq 0$ Assume that the positive direction is upward and $s(0)=0.$
a. Determine the position function, for $t \geq 0.$
b. Graph the position function on the interval [0,4].
c. At what times does the mass reach its low point the first three times?
d. At what times does the mass reach its high point the first three times?

Kevin Luu

Numerade Educator

Problem 20

Cycling distance A cyclist rides down a long straight road at a velocity (in $\mathrm{m} / \mathrm{min}$ ) given by $v(t)=400-20 t,$ for $0 \leq t \leq 10.$
a. How far does the cyclist travel in the first 5 min?
b. How far does the cyclist travel in the first 10 min?
c. How far has the cyclist traveled when her velocity is $250 \mathrm{m} / \mathrm{min} ?$

Kevin Luu

Numerade Educator

Problem 21

Flying into a headwind The velocity (in miles/hour) of an airplane flying into a headwind is given by $v(t)=30\left(16-t^{2}\right),$ for $0 \leq t \leq 3 .$ Assume that $s(0)=0.$
a. Determine and graph the position function, for $0 \leq t \leq 3.$
b. How far does the airplane travel in the first 2 hr?
c. How far has the airplane traveled at the instant its velocity reaches $400 \mathrm{mi} / \mathrm{hr} ?$

Kevin Luu

Numerade Educator

Problem 22

Day hike The velocity (in miles/hour) of a hiker walking along a straight trail is given by $v(t)=3 \sin ^{2}(\pi t / 2),$ for $0 \leq t \leq 4$ Assume that $s(0)=0.$
a. Determine and graph the position function, for $0 \leq t \leq 4.$
b. What is the distance traveled by the hiker in the first 15 min of the hike? (Hint: $\sin ^{2} t=\frac{1}{2}(1-\cos 2 t).$
c. What is the hiker's position at $t=3 ?$

Kevin Luu

Numerade Educator

Problem 23

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function $$v(t)=\left\{\begin{array}{ll} 3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45 \end{array}\right.$$, where $t$ is measured in seconds and $v$ has units of meters/second.
a. Graph the velocity function, for $0 \leq t \leq 70 .$ When is the velocity a maximum? When is the velocity zero?
b. What is the distance traveled by the automobile in the first 30 s?
c. What is the distance traveled by the automobile in the first $60 \mathrm{s} ?$
d. What is the position of the automobile when $t=75 ?$

Kevin Luu

Numerade Educator

Problem 24

Probe speed A data collection probe is dropped from a stationary balloon and it falls with a velocity (in meters/second) given by $v(t)=9.8 t,$ neglecting air resistance. After $10 \mathrm{s}$, a chute deploys and the probe immediately slows to a constant speed of $10 \mathrm{m} / \mathrm{s}$ which it maintains until it enters the ocean.
a. Graph the velocity function.
b. How far does the probe fall in the first 30 s after it is released?
c. If the probe was released from an altitude of $3 \mathrm{km},$ when does it enter the ocean?

Kevin Luu

Numerade Educator

Problem 25

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=-32, v(0)=70, s(0)=10$$

Kevin Luu

Numerade Educator

Problem 26

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=-32, v(0)=50, s(0)=0$$

Kevin Luu

Numerade Educator

Problem 27

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=-9.8, v(0)=20, s(0)=0$$

Kevin Luu

Numerade Educator

Problem 28

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=e^{-t}, v(0)=60, s(0)=40$$

Kevin Luu

Numerade Educator

Problem 29

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=-0.01 t, v(0)=10, s(0)=0$$

Kevin Luu

Numerade Educator

Problem 30

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=\frac{20}{(t+2)^{2}}, v(0)=20, s(0)=10$$

Kevin Luu

Numerade Educator

Problem 31

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=\cos 2 t, v(0)=5, s(0)=7$$

Kevin Luu

Numerade Educator

Problem 32

Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position.
$$a(t)=\frac{2 t}{\left(t^{2}+1\right)^{2}}, v(0)=0, s(0)=0$$

Kevin Luu

Numerade Educator

Problem 33

Acceleration A drag racer accelerates at $a(t)=88 \mathrm{ft} / \mathrm{s}^{2} .$ Assume that $v(0)=0$ and $s(0)=0.$
a. Determine and graph the position function, for $t \geq 0.$
b. How far does the racer travel in the first 4 seconds?
c. At this rate, how long will it take the racer to travel $\frac{1}{4} \mathrm{mi} ?$
d. How long does it take the racer to travel $300 \mathrm{ft}$ ?
e. How far has the racer traveled when it reaches a speed of $178 \mathrm{ft} / \mathrm{s} ?$

Kevin Luu

Numerade Educator

Problem 34

Deceleration A car slows down with an acceleration of $a(t)=-15 \mathrm{ft} / \mathrm{s}^{2} .$ Assume that $v(0)=60 \mathrm{ft} / \mathrm{s}$ and $s(0)=0.$
a. Determine and graph the position function, for $t \geq 0.$
b. How far does the car travel in the time it takes to come to rest?

Kevin Luu

Numerade Educator

Problem 35

Approaching a station At $t=0,$ a train approaching a station begins decelerating from a speed of $80 \mathrm{mi} / \mathrm{hr}$ according to the acceleration function $a(t)=-1280(1+8 t)^{-3},$ where $t \geq 0$ is measured in hours. How far does the train travel between $t=0$ and $t=0.2 ?$ Between $t=0.2$ and $t=0.4 ?$ The units of acceleration are $\mathrm{mi} / \mathrm{hr}^{2}.$

Kevin Luu

Numerade Educator

Problem 36

Peak oil extraction The owners of an oil reserve begin extracting oil at time $t=0 .$ Based on estimates of the reserves, suppose the projected extraction rate is given by $Q^{\prime}(t)=3 t^{2}(40-t)^{2}$ where $0 \leq t \leq 40, Q$ is measured in millions of barrels, and $t$ is measured in years.
a. When does the peak extraction rate occur?
b. How much oil is extracted in the first $10,20,$ and 30 years?
c. What is the total amount of oil extracted in 40 years?
d. Is one-fourth of the total oil extracted in the first one-fourth of the extraction period? Explain.

Kevin Luu

Numerade Educator

Problem 37

Oil production An oil refinery produces oil at a variable rate given by $$Q^{\prime}(t)=\left\{\begin{array}{ll}
800 & \text { if } 0 \leq t<30 \\
2600-60 t & \text { if } 30 \leq t<40 \\
200 & \text { if } t \geq 40
\end{array}\right.$$, where $t$ is measured in days and $Q$ is measured in barrels.
a. How many barrels are produced in the first 35 days?
b. How many barrels are produced in the first 50 days?
c. Without using integration, determine the number of barrels produced over the interval [60,80]

Kevin Luu

Numerade Educator

Problem 38

Starting with an initial value of $P(0)=55,$ the population of a prairie dog community grows at a rate of $P^{\prime}(t)=20-t / 5$ (in units of prairie dogs/month), for $0 \leq t \leq 200.$
a. What is the population 6 months later?
b. Find the population $P(t),$ for $0 \leq t \leq 200.$

Kevin Luu

Numerade Educator

Problem 39

When records were first kept $(t=0),$ the population of a rural town was 250 people. During the following years, the population grew at a rate of $P^{\prime}(t)=30(1+\sqrt{t}),$ where $t$ is measured in years.
a. What is the population after 20 years?
b. Find the population $P(t)$ at any time $t \geq 0.$

Kevin Luu

Numerade Educator

Problem 40

The population of a community of foxes is observed to fluctuate on a 10 -year cycle due to variations in the availability of prey. When population measurements began $(t=0),$ the population was 35 foxes. The growth rate in units of foxes/ year was observed to be $$P^{\prime}(t)=5+10 \sin \left(\frac{\pi t}{5}\right)$$. a. What is the population 15 years later? 35 years later? b. Find the population $P(t)$ at any time $t \geq 0.$

Willis James

Numerade Educator

Problem 41

A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of $N^{\prime}(t)=100 e^{-0.25 t}.$
a. What is the population after 20 days? After 40 days?
b. Find the population $N(t)$ at any time $t \geq 0.$

Kevin Luu

Numerade Educator

Problem 42

Endangered species The population of an endangered species changes at a rate given by $P^{\prime}(t)=30-20 t$ (individuals/year). Assume the initial population of the species is 300 individuals.
a. What is the population after 5 years?
b. When will the species become extinct?
c. How does the extinction time change if the initial population is 100 individuals? 400 individuals?

Kevin Luu

Numerade Educator

Problem 43

Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
$$C^{\prime}(x)=2000-0.5 x$$

Kevin Luu

Numerade Educator

Problem 44

Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
$$C^{\prime}(x)=200-0.05 x$$

Kevin Luu

Numerade Educator

Problem 45

Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
$$C^{\prime}(x)=300+10 x-0.01 x^{2}$$

Kevin Luu

Numerade Educator

Problem 46

Marginal cost Consider the following marginal cost functions.
a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.
b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.
$$C^{\prime}(x)=3000-x-0.001 x^{2}$$

Kevin Luu

Numerade Educator

Problem 47

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The distance traveled by an object moving along a line is the same as the displacement of the object.
b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.
c. Consider a tank that is filled and drained at a flow rate of $V^{\prime}(t)=1-t^{2} / 100(\operatorname{gal} / \min ),$ for $t \geq 0 .$ It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty.
d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from $A$ units to $2 A$ units is greater than the cost of increasing production from $2 A$ units to $3 A$ units.

Kevin Luu

Numerade Educator

Problem 48

Velocity graphs The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of $s(0)=0 .$ Determine the following:
a. The displacement between $t=0$ and $t=5$
b. The distance traveled between $t=0$ and $t=5$
c. The position at $t=5$
d. A piecewise function for $s(t)$
(FIGURE CAN'T COPY)

Kevin Luu

Numerade Educator

Problem 49

Velocity graphs The figures show velocity functions for motion along a straight line. Assume the motion begins with an initial position of $s(0)=0 .$ Determine the following:
a. The displacement between $t=0$ and $t=5$
b. The distance traveled between $t=0$ and $t=5$
c. The position at $t=5$
d. A piecewise function for $s(t)$
(FIGURE CAN'T COPY)

Kevin Luu

Numerade Educator

Problem 50

Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ______.
$$v(t)=2 t+6, \text { for } 0 \leq t \leq 8$$

Kevin Luu

Numerade Educator

Problem 51

Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ______.
$$v(t)=1-t^{2} / 16, \text { for } 0 \leq t \leq 4$$

Kevin Luu

Numerade Educator

Problem 52

Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ______.
$$v(t)=2 \sin t, \text { for } 0 \leq t \leq \pi$$

Kevin Luu

Numerade Educator

Problem 53

Equivalent constant velocity Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of ______.
$$v(t)=t\left(25-t^{2}\right)^{1 / 2}, \text { for } 0 \leq t \leq 5$$

Kevin Luu

Numerade Educator

Problem 54

Where do they meet? Kelly started at noon $(t=0)$ riding a bike from Niwot to Berthoud, a distance of $20 \mathrm{km},$ with velocity $v(t)=15 /(t+1)^{2}$ (decreasing because of fatigue). Sandy started at noon $(t=0)$ riding a bike in the opposite direction from Berthoud to Niwot with velocity $u(t)=20 /(t+1)^{2}$ (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.
a. Make a graph of Kelly's distance from Niwot as a function of time.
b. Make a graph of Sandy's distance from Berthoud as a function of time.
c. How far has each person traveled when they meet? When do they meet?
d. More generally, if the riders' speeds are $v(t)=A /(t+1)^{2}$ and $u(t)=B /(t+1)^{2}$ and the distance between the towns is $D,$ what conditions on $A, B,$ and $D$ must be met to ensure that the riders will pass each other?
e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

Kevin Luu

Numerade Educator

Problem 55

Bike race Theo and Sasha start at the same place on a straight road riding bikes with the following velocities (measured in $\mathrm{mi} / \mathrm{hr}$ ): Theo: $v_{T}(t)=10,$ for $t \geq 0$
Sasha: $v_{S}(t)=15 t,$ for $0 \leq t \leq 1$ and $v_{S}(t)=15,$ for $t>1.$ a. Graph the velocity functions for both riders.
b. If the riders ride for 1 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a).
c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a).
d. Which rider arrives first at the $10-, 15$ -, and 20 -mile markers of the race? Interpret your answer geometrically using the graphs of part (a).
e. Suppose Sasha gives Theo a head start of 0.2 mi and the riders ride for $20 \mathrm{mi}$. Who wins the race?
f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

Kevin Luu

Numerade Educator

Problem 56

Two runners At noon $(t=0),$ Alicia starts running along a long straight road at $4 \mathrm{mi} / \mathrm{hr}$. Her velocity decreases according to the function $v(t)=4 /(t+1),$ for $t \geq 0 .$ At noon, Boris also starts running along the same road with a 2 -mi head start on Alicia; his velocity is given by $u(t)=2 /(t+1),$ for $t \geq 0.$
a. Find the position functions for Alicia and Boris, where $s=0$ corresponds to Alicia's starting point.
b. When, if ever, does Alicia overtake Boris?

Kevin Luu

Numerade Educator

Problem 57

Running in a wind A strong west wind blows across a circular running track. Abe and Bess start at the south end of the track and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of miles/hour) given by $u(\varphi)=3-2 \cos \varphi$ and Bess runs with a speed given by $v(\theta)=3+2 \cos \theta,$ where $\varphi$ and $\theta$ are the central angles of the runners. (FIGURE CAN'T COPY)
a. Graph the speed functions $u$ and $v,$ and explain why they describe the runners' speeds (in light of the wind).
b. Compute each runner's average speed (over one lap) with respect to the central angle.
c. Challenge: If the track has a radius of $\frac{1}{10} \mathrm{mi}$, how long does it take each runner to complete one lap and who wins the race?

Kevin Luu

Numerade Educator

Problem 58

Filling a tank A $200-\mathrm{L}$ cistern is empty when water begins flowing into it (at $t=0$ ) at a rate (in liters/minute) given by $Q^{\prime}(t)=3 \sqrt{t}.$
a. How much water flows into the cistern in 1 hour?
b. Find and graph the function that gives the amount of water in the tank at any time $t \geq 0.$
c. When will the tank be full?

Kevin Luu

Numerade Educator

Problem 59

Depletion of natural resources Suppose that $r(t)=r_{0} e^{-k t}$ with $k \geq 0,$ is the rate at which a nation extracts oil, where $r_{0}=10^{7}$ barrels / yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is $2 \times 10^{9}$ barrels.
a. Find $Q(t),$ the total amount of oil extracted by the nation after $t$ years.
b. Evaluate $\lim _{t \rightarrow \infty} Q(t)$ and explain the meaning of this limit.
c. Find the minimum decay constant $k$ for which the total oil reserves will last forever.
d. Suppose $r_{0}=2 \times 10^{7}$ barrels/yr and the decay constant $k$ is the minimum value found in part (c). How long will the total oil reserves last?

Kevin Luu

Numerade Educator

Problem 60

Snowplow problem With snow on the ground and falling at a constant rate, a snowplow began plowing down a long straight road at noon. The plow traveled twice as far in the first hour as it did in the second hour. At what time did the snow start falling? Assume the plowing rate is inversely proportional to the depth of the snow.

Kevin Luu

Numerade Educator

Problem 61

Filling a reservoir A reservoir with a capacity of $2500 \mathrm{m}^{3}$ is filled with a single inflow pipe. The reservoir is empty when the inflow pipe is opened at $t=0 .$ Letting $Q(t)$ be the amount of water in the reservoir at time $t$, the flow rate of water into the reservoir (in $\mathrm{m}^{3} / \mathrm{hr}$ ) oscillates on a 24 -hr cycle (see figure) and is given by $$Q^{\prime}(t)=20\left[1+\cos \left(\frac{\pi t}{12}\right)\right]$$. (GRAPH CAN'T COPY) a. How much water flows into the reservoir in the first 2 hr?
b. Find and graph the function that gives the amount of water in the reservoir over the interval $[0, t],$ where $t \geq 0.$ c. When is the reservoir full?

Kevin Luu

Numerade Educator

Problem 62

Blood flow A typical human heart pumps 20 mL of blood with each stroke (stroke volume). Assuming a heart rate of
60 beats / min, a reasonable model for the outflow rate of the heart is $V^{\prime}(t)=20(1+\sin (2 \pi t)),$ where $V(t)$ is the amount of blood (in milliliters) pumped over the interval $[0, t], V(0)=0,$ and $t$ is measured in seconds.
a. Graph the outflow rate function.
b. Verify that the amount of blood pumped over a one-second interval is $20 \mathrm{mL}.$
c. Find the function that gives the total blood pumped between $t=0$ and a future time $t>0.$
d. What is the cardiac output over a period of 1 min? (Use calculus, then check your answer with algebra.)

Kevin Luu

Numerade Educator

Problem 63

Air flow in the lungs A reasonable model (with different parameters for different people $)$ for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi V_{0}}{10} \sin \left(\frac{\pi t}{5}\right)$$, where $V(t)$ is the volume of air in the lungs at time $t \geq 0,$ measured in liters, $t$ is measured in seconds, and $V_{0}$ is the capacity of the lungs. The time $t=0$ corresponds to a time at which the lungs are full and exhalation begins.
a. Graph the flow rate function with $V_{0}=10 \mathrm{L}$
b. Find and graph the function $V$, assuming that $V(0)=V_{0}=10 \mathrm{L}$
c. What is the breathing rate in breaths/minute?

Kevin Luu

Numerade Educator

Problem 64

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period $P$. Consider the growth rate function $$N^{\prime}(t)=A \sin \left(\frac{2 \pi t}{P}\right)+r$$. where $A$ and $r$ are constants with units of individuals/year. A species becomes extinct if its population ever reaches 0 after $t=0.$
a. Suppose $P=10, A=20,$ and $r=0 .$ If the initial population is $N(0)=10,$ does the population ever become extinct? Explain.
b. Suppose $P=10, A=20,$ and $r=0 .$ If the initial population is $N(0)=100$, does the population ever become extinct? Explain.
c. Suppose $P=10, A=50,$ and $r=5 .$ If the initial population is $N(0)=10,$ does the population ever become extinct? Explain.
d. Suppose $P=10, A=50,$ and $r=-5 .$ Find the initial population $N(0)$ needed to ensure that the population never becomes extinct.

Kevin Luu

Numerade Educator

Problem 65

Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal $=4184$ J. One hour of walking consumes roughly $10^{6} \mathrm{J}$, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts $(\mathrm{W} ; 1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})$ Other useful units of power are kilowatts $\left(1 \mathrm{kW}=10^{3} \mathrm{W}\right)$ and megawatts $\left(1 \mathrm{MW}=10^{6} \mathrm{W}\right)$. If energy is used at a rate of $1 \mathrm{kW}$ for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is $3.6 \times 10^{6} \mathrm{J}$
Suppose the power function of a large city over a 24 -hr period is given by $$P(t)=E^{\prime}(t)=300-200 \sin \left(\frac{\pi t}{12}\right)$$, where $P$ is measured in megawatts and $t=0$ corresponds to 6: 00 p.m. (see figure). (GRAPH CAN'T COPY) a. How much energy is consumed by this city in a typical 24 -hr period? Express the answer in megawatt-hours and in joules. b. Burning $1 \mathrm{kg}$ of coal produces about $450 \mathrm{kWh}$ of energy. How many kg of coal are required to meet the energy needs of the city for 1 day? For 1 year? c. Fission of $1 \mathrm{g}$ of uranium- 235 (U- 235 ) produces about $16,000 \mathrm{kWh}$ of energy. How many grams of uranium are needed to meet the energy needs of the city for 1 day? For 1 year? d. A typical wind turbine can generate electricity at a rate of about $200 \mathrm{kW}$. Approximately how many wind turbines are needed to meet the average energy needs of the city?

Kevin Luu

Numerade Educator

Problem 66

Variable gravity At Earth's surface the acceleration due to gravity is approximately $g=9.8 \mathrm{m} / \mathrm{s}^{2}$ (with local variations). However, the acceleration decreases with distance from the surface according to Newton's law of gravitation. At a distance of $y$ meters from Earth's surface, the acceleration is given by $$a(y)=-\frac{g}{(1+y / R)^{2}}$$, where $R=6.4 \times 10^{6} \mathrm{m}$ is the radius of Earth.
a. Suppose a projectile is launched upward with an initial velocity of $v_{0} \mathrm{m} / \mathrm{s} .$ Let $v(t)$ be its velocity and $y(t)$ its height (in meters) above the surface $t$ seconds after the launch. Neglecting forces such as air resistance, explain why $\frac{d v}{d t}=a(y)$ and $\frac{d y}{d t}=v(t).$
b. Use the Chain Rule to show that $\frac{d v}{d t}=\frac{1}{2} \frac{d}{d y}\left(v^{2}\right).$
c. Show that the equation of motion for the projectile is $\frac{1}{2} \frac{d}{d y}\left(v^{2}\right)=a(y),$ where $a(y)$ is given previously.

James Kiss

Numerade Educator