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Chapter 3

Differentiation Rules

Educators

AL
WZ
+ 12 more educators

Problem 1

If $ V $ is the volume of a cube with edge length $ x $ and the cube expands as time passes, find $ dV/dt $ in terms of $ dx/dt. $

Chris T.
Numerade Educator

Problem 2

(a) If $ A $ is the area of a circle with radius $ r $ and the circle expands as time passes, find $ dA/dt $ in terms of $ dr/dt. $
(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spills increases at a constant rate of $ 1 m/s $, how fast is the area of the spill increasing when the radius is $ 30 m? $

Chris T.
Numerade Educator

Problem 3

Each side of a square is increasing at a rate of $ 6 cm/s. $ At what rate is the area of the square increasing when the area of the square is 16 $ cm^2? $

Sam L.
Numerade Educator

Problem 4

The length of a rectangle is increasing at a rate of $ 8 cm/s $ and its width is increasing at a rate of $ 3 cm/s. $ When the length is $ 20 cm $ and the width is $ 10 cm, $ how fast is the area of the rectangle increasing?

Sam L.
Numerade Educator

Problem 5

A cylindrical tank with radius $ 5 m $ is being filled with water at a rate of $ 3 m^3/min. $ How fast is the height of the water increasing?

Sam L.
Numerade Educator

Problem 6

The radius of a sphere is increasing at a rate of $ 4 mm/s. $ How fast is the volume increasing when the diameter is $ 80 mm? $

Sam L.
Numerade Educator

Problem 7

The radius of a spherical ball is increasing at a rate of $ 2 cm/min. $ At what rate is the surface area of the ball increasing when the radius is $ 8 cm? $

Chris T.
Numerade Educator

Problem 8

The area of a triangle with sides of length $ a $ and $ b $ and contained angle $ \theta $ is
$ A = \frac {1}{2}ab \sin \theta $
(a) If $ a = 2 cm, b = 3 cm, $ and $ \theta $ increases at a rate of $ 0.2 rad/min, $ how fast is the area increasing when $ \theta = \pi/3? $
(b) If $ a = 2 cm, b $ increases at a rate of $ 1.5 cm/min, $ and $ \theta $ increases at rate of $ 0.2 rad/min, $ how fast is the area increasing when $ b = 3 cm $ and $ \theta = \pi/3? $
(c) If $ a $ increases at a rate of $ 1.5 cm, b $ increases at a rate of $ 1.5 cm/min, $ and $ \theta $ increases at a rate of $ 0.2 rad/min, $ how fast is the area increasing when $ a = 2 cm, b = 3 cm, $ and $ \theta = \pi/3? $

Linda H.
Numerade Educator

Problem 9

Suppose $ y = \sqrt {2x + 1}, $ where $ x $ and $ y $ are function of $ t $,
(a) If $ dx/dt = 3, $ find $ dy/dt $ when $ x = 4. $
(b) If $ dy/dt = 5, $ find $ dx/dt $ when $ x = 12. $

Carson M.
Numerade Educator

Problem 10

Suppose $ 4x^2 + 9y^2 = 36, $ where $ x $ and $ y $ are functions of $ t $.
(a) If $ dy/dt = \frac {1}{3}, $ find $ dx/dt $ when $ x = 2 $ and $ y = \frac {2}{3} \sqrt 5. $
(b) If $ dx/dt = 3, $ find $ dy/dt $ when $ x = -2 $ and $ y = \frac {2}{3} \sqrt 5. $

Chris T.
Numerade Educator

Problem 11

If $ x^2 + y^2 + z^2 = 9, dx/dt = 5, $ and $ dy/dt = 4, $ find $ dz/dt $ when $ (x, y, z) = (2, 2, 1). $

Carson M.
Numerade Educator

Problem 12

A particle is moving along a hyperbola $ xy = 8. $ As it reaches the point (4, 2), the $ y- $ coordinate is decreasing at a rate of $ 3 cm/s. $ How fast is the $ x- $ coordinate of the point changing at that instant?

Chris T.
Numerade Educator

Problem 13

A plane flying horizontally at an altitude of $ 1 mi $ and a speed of $ 500 mi/h $ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $ 2 mi $ away from the station.
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time $ t. $
(d) Write an equation that relates the quantities.
(e) Finish solving the problem.

Chris T.
Numerade Educator

Problem 14

If a snowball melts so that its surface area decreases at a rate of $ 1cm^2/min, $ find the rate at which the diameter decreases when the diameter is $ 10 cm. $
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time $ t. $
(d) Write an equation that relates the quantities.
(e) Finish solving the problem.

WZ
Wen Z.
Numerade Educator

Problem 15

A street light mounted at the top of a $ 15-ft-tall $ pole. A man $ 6 ft $ tall walks away from the pole with a speed of $ 5 ft/s $ along a straight path. How fast is the tip of his shadow moving when he is $ 40 ft $ from the pole?
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time $ t. $
(d) Write an equation that relates the quantities.
(e) Finish solving the problem.

WZ
Wen Z.
Numerade Educator

Problem 16

At noon, ship $ A $ is $ 150 km $ west of ship $ B. $ Ship $ A $ is sailing east at $ 35 km/h $ and ship $ B $ is sailing north at $ 25 km/h. $ How fast is the distance between the ships changing at $ 4:00 PM? $
(a) What quantities are given in the problem?
(b) What is the unknown?
(c) Draw a picture of the situation for any time $ t. $
(d) Write an equation that relates the quantities.
(e) Finish solving the problem.

Carson M.
Numerade Educator

Problem 17

Two cars start moving from the same from the same point. One travels south at $ 60 mi/h $ and the other travels west at $ 25 mi/h. $ At what rate is the distance between the cars increasing two hours later?

Amrita B.
Numerade Educator

Problem 18

A spotlight on the ground shines on a wall $ 12 m $ away. If a man $ 2 m $ tall from the spotlight towards the building at a speed of $ 1.6 m/s, $ how fast is the length of his shadow on the building decreasing when he is $ 4 m $ from the building?

Chris T.
Numerade Educator

Problem 19

A man stars walking north at $ 4 ft/s $ from a point $ P. $ Five minutes later a woman starts walking south at $ 5 ft/s $ from a point $ 500 ft $ due east $ P. $ At what rate are the people moving apart $ 15 min $ after the woman starts walking?

Chris T.
Numerade Educator

Problem 20

A baseball diamond is a square with side $ 90 ft. $ A batter hits the ball and runs toward first base with a speed of $ 24 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 hits distance from third base increasing at the same moment?

Chris T.
Numerade Educator

Problem 21

The altitude of a triangle is increasing at a rate of $ 1 cm/min $ while the area of the triangle is increasing at a rate of $ 2 cm^2/min. $ At what rate is the base of the triangle changing when the altitude is $ 10 cm $ and the area is $ 100 cm^2? $

Amrita B.
Numerade Educator

Problem 22

The altitude of a triangle is increasing at a rate of $ 1 cm/min $ while the area of the triangle is increasing at a rate of $ 2 cm^2/min. $ At what rate is the base of the triangle changing when the altitude is $ 10 cm $ and the area is $ 100 cm^2? $

Suman Saurav T.
Numerade Educator

Problem 23

At noon, ship $ A $ is $ 100 km $ west of ship $ B. $ Ship $ A $ is sailing south at $ 35 km/h $ and ship $ B $ is sailing north at $ 25 km/h. $ How fast is the distance between the ships changing at $ 4:00 PM? $

Amrita B.
Numerade Educator

Problem 24

A particle moves along the curve $ y = 2 \sin (\pi x/2). $ As the particle passes through the point $ (\frac {1}{3}, 1). $ its $ x- $ coordinate increases at a rate of $ \sqrt {10} cm/s. $ How fast is the distance from the particle to the origin changing at this instant?

Amrita B.
Numerade Educator

Problem 25

Water is leaking out of an inverted conical tank at a rate of $ 10,000 cm^3/min $ at the same time that water is being pumped into the tank at a constant rate. The tank has height $ 6 m $ and the diameter at the top is $ 4 m. $ If the water level is rising at a rate of $ 20 cm/min $ when the height of the water is $ 2 m, $ find the rate at which water is being pumped into the tank.

Amrita B.
Numerade Educator

Problem 26

A trough is $ 10 ft $ long and its ends have the shape of isosceles triangles that are $ 3 ft $ across at the top and have a height of $ 1 ft. $ If the trough is being filled with water at a rate of $ 12 ft^3/ min, $ how fast is the water level rising when the water is $ 6 inches $ deep?

Amrita B.
Numerade Educator

Problem 27

A water trough is $ 10 m $ long and a cross-section has the shape of an isosceles trapezoid that is $ 30 cm $ wide at the bottom, $ 80 cm $ wide at the top, and has height $ 50 cm, $ If the trough is being filled with water at the rate $ 0.2 m^3/min, $ how fast is the water level rising when the water is $ 30 cm $ deep?

Carson M.
Numerade Educator

Problem 28

A swimming pool is 20 $\mathrm{ft}$ wide, 40 $\mathrm{ft}$ long, 3 $\mathrm{ft}$ deep at the
shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.8 $\mathrm{ft}^{3} / \mathrm{min}$ , how fast is the water level rising when
the depth at the deepest point is 5 $\mathrm{ft}$ ?

Amrita B.
Numerade Educator

Problem 29

Gravel is being dumped from a conveyor belt at a rate of $ 30 ft^3/min, $ and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is $ 10 ft $ high?

Amrita B.
Numerade Educator

Problem 30

A kite $ 100 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 $ 200 ft $ of string has been let out?

Amrita B.
Numerade Educator

Problem 31

The sides of an equilateral triangle are increasing at a rate of $ 10 cm/min. $ At what rate is the area of the triangle increasing when the sides are $ 30 cm $ long?

Anthony H.
Numerade Educator

Problem 32

How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall?

Mutahar M.
Numerade Educator

Problem 33

The top of a ladder slides down a vertical wall at a rate of $ 0.15 m/s. $ At the moment when the bottom of the ladder is $ 3 m $ from the wall, it slides away from the wall at a rate of $ 0.2 m/s. $ How long is the ladder?

Carson M.
Numerade Educator

Problem 34

According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? is the model appropriate for small values of $ y? $

Chris T.
Numerade Educator

Problem 35

If the minute hand of clock has length $ r $ (in centimeters), find the rate at which it sweeps out area as a function of $ r, $

Suman Saurav T.
Numerade Educator

Problem 36

A faucet is filling a hemispherical basin of diameter $60 \mathrm{~cm}$ with water at a rate of $2 \mathrm{~L} / \mathrm{min}$. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: $1 \mathrm{~L}$ is $1000 \mathrm{~cm}^{3}$. The volume of the portion of a sphere with radius $r$ from the bottom to a height $h$ is $V=\pi\left(r h^{2}-\frac{1}{3} h^{3}\right),$ as we will show in Chapter $6 .$

Linda H.
Numerade Educator

Problem 37

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure $ P $ and volume $ V $ satisfy the equation $ PV = C, $ where $ C $ is constant, the pressure is $ 150 kPa, $ and the pressure is increasing at a rate of $ 20 kPa/min. $ At what rate is the volume decreasing at this instant?

Chris T.
Numerade Educator

Problem 38

When air expands adiabatically (without gaining or losing heat), its pressure $ P $ and volume $ V $ are related by the equation $ PV^{1.4} = C, $ where $ C $ is a constant. Suppose that at a certain instant the volume is $ 400 cm^3 $ and the pressure is $ 80 kPa $ and is decreasing at a rate of $ 10 kPa/min. $ At what rate is the volume increasing at this instant?

Amrita B.
Numerade Educator

Problem 39

If two resistors with resistances $ R_1 $ and $ R_2 $ are connected in parallel, as in the figure, then the total resistance $ R, $ measured in ohms $ (\Omega), $ is given by
$ \frac {1}{R} = \frac {1}{R_1} + \frac {1}{R_2} $
If $ R_1 $ and $ R_2 $ are increasing at rates of $ 0.3 \Omega/s $ and $ 0.2 \Omega/s, $ respectively, how fast is $ R $ changing when $ R_1 = 80 \Omega $ and $ R_2 = 100 \Omega? $

Amrita B.
Numerade Educator

Problem 40

Brain weight $ B $ as a function of body weight $ W $ in fish has been modeled by the power function $ B = 0.007W^{2/3}, $ where $ B $ and $ W $ are measured in grams. A model for body weight as a function of body length $ L $ (measured in centimeters) is $ W = 0. 12L^{2.53}. $ If, over 10 million years, the average length of a certain species of fish evolved from $ 15 cm $ to $ 20 cm $ at a constant rate, how fast was this species' brain growing when the average length was $ 18 cm? $

Amrita B.
Numerade Educator

Problem 41

Two sides of a triangle have lengths $ 12 m $ and $ 15 m. $ The angle between them is increasing at a rate of $ 2^o/min. $ How fast is the length of the third side increasing when the angle between the side of fixed length is $ 60^o? $

Bobby B.
University of North Texas

Problem 42

Two carts, $\mathrm{A}$ and $\mathrm{B},$ are connected by a rope $39 \mathrm{ft}$ long that passes over a pulley $P$ (see the figure). The point $Q$ is on the floor $12 \mathrm{ft}$ directly beneath $P$ and between the carts. Cart A is being pulled away from $Q$ at a speed of $2 \mathrm{ft} / \mathrm{s}$. How fast is cart B moving toward $Q$ at the instant when cart A is $5 \mathrm{ft}$ from $Q ?$

EI
Eric I.
Numerade Educator

Problem 43

A television camera is positioned $ 4000 ft $ from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the speed is $ 600 ft/s $ when it has risen $ 3000 ft. $
(a) How fast is the distance from the television camera to the rocket changing at that moment?
(b) If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment?

Carolyn B.
Numerade Educator

Problem 44

A lighthouse is located on a small island $ 3 km $ away from the nearest point $ P $ on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from $ P? $

Mutahar M.
Numerade Educator

Problem 45

A plane flies horizontally at an altitude of $ 5 km $ and passes directly over a tracking telescope on the ground. When the angle of elevation is $ \pi/3, $ this angle is decreasing at a rate of $ \pi/6 rad/min. $ How fast is the plane traveling at that time?

Vipin B.
Numerade Educator

Problem 46

A Ferris wheel with a radius of $ 10 m $ is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is $ 16 m $ above the ground level?

Carson M.
Numerade Educator

Problem 47

A plane flying with a constant speed of $ 300 km/h $ passes over a ground radar station at an altitude of $ 1 km $ and climbs at an angle of $ 30^o. $ At what rate is the distance from the plane to the radar station increasing a minute later?

Mutahar M.
Numerade Educator

Problem 48

Two people start from the same point. One walks east at $ 3 mi/h $ and the other walks northeast at $ 2 mi/h. $ How fast is the distance between the people changing after 15 minutes?

Amrita B.
Numerade Educator

Problem 49

A runner sprints around a circular track of radius $ 100 m $ at a constant speed of $ 7 m/s. $ The runner's friend is standing at a distance $ 200 m $ from the center of the track. How fast is the distance between the friend changing when the distance between them is $ 200 m? $

Carson M.
Numerade Educator

Problem 50

The minute hand on a watch is $ 8 mm $ long and the hour hand is $ 4 mm $ long. How fast is the distance between the tips of the hands changing at one o'clock?

Amrita B.
Numerade Educator