Chapter 24, Applications of the Derivative Video Solutions, Basic Technical Mathematics with Calculus

Section 4

Related Rates

Problem 1

Make the given changes in the indicated examples of this section and then solve the resulting problems.
In Example $1,$ change 0.006 to 0.012.

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Make the given changes in the indicated examples of this section and then solve the resulting problems.
In Example $3,$ change "volume" to "surface area," change $\mathrm{ft}^{3} / \mathrm{min}$ to $\mathrm{ft}^{2} / \mathrm{min}$.

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Problem 3

Assume that all variables are implicit functions of time t. Find the indicated rates.
$y=5 x^{2}-4 x ; d x / d t=0.5$ when $x=5 ;$ find $d y / d t$.

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Problem 4

Assume that all variables are implicit functions of time t. Find the indicated rates.
$y=\sqrt{9-2 x^{2}} ; d y / d t=3$ when $x=2$ and $y=1 ;$ find $d x / d t$.

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Problem 5

Assume that all variables are implicit functions of time t. Find the indicated rates.
$x^{2}+3 y^{2}+2 y=10 ; d x / d t=2$ when $x=3$ and $y=-1 ;$ find $d y / d t$.

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Problem 6

Assume that all variables are implicit functions of time t. Find the indicated rates.
$z=2 x^{2}-3 x y ; d x / d t=-2, d y / d t=3$ when $x=1$ and $y=4$ find $d z / d t$.

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Problem 7

Solve the problems in related rates.
The velocity $v$ (in $\mathrm{ft} / \mathrm{s}$ ) of a pulse traveling in a certain string is a function of the tension $T(\text { in } \mathrm{lb} \text { ) in the string given by } v=18 \sqrt{T}$. Find $d v / d t$ if $d T / d t=0.20 \mathrm{lb} / \mathrm{s}$ when $T=25 \mathrm{lb}$.

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Problem 8

Solve the problems in related rates.
The force $F$ (in $1 \mathrm{b}$ ) on the blade of a certain wind generator as a function of the wind velocity $v$ (in $\mathrm{ft} / \mathrm{s}$ ) is given by $F=0.0056 v^{2}$. Find $d F / d t$ if $d v / d t=0.75 \mathrm{ft} / \mathrm{s}^{2}$ when $v=28 \mathrm{ft} / \mathrm{s}$.

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Problem 9

Solve the problems in related rates.
The electric resistance $R$ (in $\Omega$ ) of a certain resistor as a function of the temperature $T\left(\text { in }^{\circ} \mathrm{C}\right)$ is $R=4.000+0.003 T^{2} .$ If the temperature is increasing at the rate of $0.100^{\circ} \mathrm{C} / \mathrm{s}$, find how fast the resistance changes when $T=150^{\circ} \mathrm{C}$.

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Problem 10

Solve the problems in related rates.
The kinetic energy $K$ (in $J$ ) of an object is given by $K=\frac{1}{2} m v^{2}$, where $m$ is the mass (in $\mathrm{kg}$ ) of the object and $v$ is its velocity. If a $250-\mathrm{kg}$ wrecking ball accelerates at $5.00 \mathrm{m} / \mathrm{s}^{2}$, how fast is the kinetic energy changing when $v=30.0 \mathrm{m} / \mathrm{s} ?$ (Hint: $\operatorname{acceleration}=\frac{d v}{d t}.$)

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Problem 11

Solve the problems in related rates.
The length $L$ (in in.) of a pendulum is slowly decreasing at the rate of 0.100 in $/$ s. How fast is the period $T$ (in s) of the pendulum changing when $L=16.0$ in., if the equation relating the period and length is $T=\pi \sqrt{L / 96} ?$

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Problem 12

Solve the problems in related rates.
The voltage $V$ that produces a current $I$ (in $A$ ) in a wire of radius $r$ (in in.) is $V=0.030 I / r^{2} .$ If the current increases at $0.020 \mathrm{A} / \mathrm{s}$ in a wire of 0.040 in. radius, find the rate at which the voltage is increasing.

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Problem 13

Solve the problems in related rates.
A plane flying at an altitude of $2.0 \mathrm{mi}$ is at a direct distance $D=\sqrt{4.0+x^{2}}$ from an airport control tower, where $x$ is the horizontal distance to the tower. If the plane's speed is $350 \mathrm{mi} / \mathrm{h}$, how fast is $D$ changing when $x=6.2$ mi? See Fig. 24.24.

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Problem 14

Solve the problems in related rates.
A variable resistor $R$ and an $8-\Omega$ resistor in parallel have a combined resistance $R_{T}$ given by $R_{T}=\frac{8 R}{8+R}$. If $R$ is changing at $0.30 \Omega /$ min, find the rate at which $R_{T}$ is changing when $R=6.0 \Omega$.

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Problem 15

Solve the problems in related rates.
The radius $r$ (in $\mathrm{m}$ ) of a ring of a certain holograph (an image produced without using a lens) is given by $r=\sqrt{0.4 \lambda},$ where $\lambda$ is the wavelength of the light being used. If $\lambda$ is changing at the rate of $0.10 \times 10^{-7} \mathrm{m} / \mathrm{s}$ when $\lambda=6.0 \times 10^{-7} \mathrm{m},$ find the rate at which $r$ is changing.

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Problem 16

Solve the problems in related rates.
An Earth satellite moves in a path that can be described by $\frac{x^{2}}{2.80 \times 10^{7}}+\frac{y^{2}}{2.76 \times 10^{7}}=1,$ where $x$ and $y$ are in miles. If $d x / d t=7750 \mathrm{mi} / \mathrm{h}$ for $x=2020 \mathrm{mi}$ and $y>0,$ find $d y / d t$.

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Problem 17

Solve the problems in related rates.
The magnetic field $B$ due to a magnet of length $l$ at a distance $r$ is given by $B=\frac{k}{\left[r^{2}+(l / 2)^{2}\right]^{3 / 2}},$ where $k$ is a constant for a given magnet. Find the expression for $\frac{d B}{d t}$ in terms of $\frac{d r}{d t}$.

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Problem 18

Solve the problems in related rates.
An approximate relationship between the pressure $p$ and volume $v$ of the vapor in a diesel engine cylinder is $p v^{1.4}=k,$ where $k$ is a constant. At a certain instant, $p=4200 \mathrm{kPa}, v=75 \mathrm{cm}^{3},$ and the volume is increasing at the rate of $850 \mathrm{cm}^{3} / \mathrm{s}$. At what rate is the pressure changing at this instant?

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Problem 19

Solve the problems in related rates.
A swimming pool with a rectangular surface $18.0 \mathrm{m}$ long and $12.0 \mathrm{m}$ wide is being filled at the rate of $0.80 \mathrm{m}^{3} / \mathrm{min} .$ At one end it is $1.0 \mathrm{m}$ deep, and at the other end it is $2.5 \mathrm{m}$ deep, with a constant slope between ends. How fast is the height of water rising when the depth of water at the deep end is $1.0 \mathrm{m} ?$

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Problem 20

Solve the problems in related rates.
An engine cylinder $15.0 \mathrm{cm}$ deep is being bored such that the radius increases by $0.100 \mathrm{mm} / \mathrm{min} .$ How fast is the volume $V$ of the cylinder changing when the diameter is $9.50 \mathrm{cm} ?$

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Problem 21

Solve the problems in related rates.
Fatty deposits have decreased the circular cross-sectional opening of a person's artery. A test drug reduces these deposits such that the radius of the opening increases at the rate of $0.020 \mathrm{mm} /$ month. Find the rate at which the area of the opening increases when $r=1.2 \mathrm{mm}$.

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Problem 22

Solve the problems in related rates.
A rectangular image 4.00 in. high on a computer screen is widening at the rate of 0.25 in./s. Find the rate at which the diagonal is increasing when the width is 6.50 in.

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Problem 23

Solve the problems in related rates.
A metal cube dissolves in acid such that an edge of the cube decreases by $0.50 \mathrm{mm} / \mathrm{min}$. How fast is the volume of the cube changing when the edge is $8.20 \mathrm{mm} ?$

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Problem 24

Solve the problems in related rates.
A metal sphere is placed in seawater to study the corrosive effect of seawater. If the surface area decreases at $35 \mathrm{cm}^{2} /$ year due to corrosion, how fast is the radius changing when it is $12 \mathrm{cm} ?$

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Problem 25

Solve the problems in related rates.
A uniform layer of ice covers a spherical water-storage tank. As the ice melts, the volume $V$ of ice decreases at a rate that varies directly as the surface area $A$. Show that the outside radius decreases at a constant rate.

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Problem 26

Solve the problems in related rates.
A light in a garage is $9.50 \mathrm{ft}$ above the floor and $12.0 \mathrm{ft}$ behind the door. If the garage door descends vertically at $1.50 \mathrm{ft} / \mathrm{s}$, how fast is the door's shadow moving toward the garage when the door is 2.00 ft above the floor?

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Problem 27

Solve the problems in related rates.
One statement of Boyle's law is that the pressure of a gas varies inversely as the volume for constant temperature. If a certain gas occupies $650 \mathrm{cm}^{3}$ when the pressure is $230 \mathrm{kPa}$ and the volume is increasing at the rate of $20.0 \mathrm{cm}^{3} / \mathrm{min}$, how fast is the pressure changing when the volume is $810 \mathrm{cm}^{3} ?$

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Problem 28

Solve the problems in related rates.
The tuning frequency $f$ of an electronic tuner is inversely proportional to the square root of the capacitance $C$ in the circuit. If $f=920 \mathrm{kHz}$ for $C=3.5 \mathrm{pF},$ find how fast $f$ is changing at this frequency if $d C / d t=0.3 \mathrm{pF} / \mathrm{s}$.

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Problem 29

Solve the problems in related rates.
The shadow of a $24-\mathrm{m}$ high building is increasing at the rate of $18 \mathrm{cm} / \mathrm{min}$ when the shadow is $18 \mathrm{m}$ long. How fast is the distance from the top of the building to the end of the shadow increasing?

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Problem 30

Solve the problems in related rates.
The acceleration due to the gravity $g$ on a spacecraft is inversely proportional to its distance from the center of Earth. At the surface of Earth, $g=32.2 \mathrm{ft} / \mathrm{s}^{2}$. Given that the radius of Earth is 3960 mi, how fast is $g$ changing on a spacecraft approaching Earth at $4500 \mathrm{ft} / \mathrm{s}$ at a distance of $25,500 \mathrm{mi}$ from the surface?

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Problem 31

Solve the problems in related rates.
The intensity $I$ of heat varies directly as the strength of the source and inversely as the square of the distance from the source. If an object approaches a heated object of strength 8.00 units at the rate of $50.0 \mathrm{cm} / \mathrm{s},$ how fast is the intensity changing when it is $100 \mathrm{cm}$ from the source?

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Problem 32

Solve the problems in related rates.
The speed of sound $v$ (in $\mathrm{m} / \mathrm{s}$ ) is $v=331 \sqrt{T / 273},$ where $T$ is the temperature (in K). If the temperature is $303 \mathrm{K}\left(30^{\circ} \mathrm{C}\right)$ and is rising at $2.0^{\circ} \mathrm{C} / \mathrm{h}$, how fast is the speed of sound rising?

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Problem 33

Solve the problems in related rates.
As a space shuttle moves into space, an astronaut's weight decreases. An astronaut weighing $650 \mathrm{N}$ at sea level has a weight of $w=650\left(\frac{6400}{6400+h}\right)$ at $h$ kilometers above sea level. If the shuttle is moving away from Earth at $6.0 \mathrm{km} / \mathrm{s}$, at what rate is $w$ changing when $h=1200 \mathrm{km} ?$

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Problem 34

Solve the problems in related rates.
The oil reservoir for the lubricating mechanism of a machine is in the shape of an inverted pyramid. It is being filled at the rate of $8.00 \mathrm{cm}^{3} / \mathrm{s}$ and the top surface is increasing at the rate of $6.00 \mathrm{cm}^{2} / \mathrm{s} .$ When the depth of oil is $6.50 \mathrm{cm}$ and the top surface area is $22.5 \mathrm{cm}^{2},$ how fast is the level increasing? (Hint: The volume of a pyramid is given by $V=\frac{1}{3} B h,$ where $B$ is the area of the base and $h$ is the height).

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Problem 35

Solve the problems in related rates.
Coffee is draining through a conical filter into a coffee pot at the rate of $18.0 \mathrm{cm} .^{3} / \mathrm{min}$. If the filter is $15.0 \mathrm{cm}$ in diameter and $15.0 \mathrm{cm}$ deep, how fast is the level of coffee in the filter changing when the depth is $10.0 \mathrm{cm} ?$ (Hint: Use $V=\frac{1}{3} \pi r^{2} h .$ )

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Problem 36

Solve the problems in related rates.
The top of a ladder $4.00 \mathrm{m}$ long is slipping down a vertical wall at a constant rate of $0.75 \mathrm{m} / \mathrm{s} .$ How fast
is the bottom of the ladder moving along the ground away from the wall when it is $2.50 \mathrm{m}$ from the wall? See Fig. 24.25.

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Problem 37

Solve the problems in related rates.
A supersonic jet leaves an airfield traveling due east at $1600 \mathrm{mi} / \mathrm{h}$. A second jet leaves the same airfield at the same time and travels $1800 \mathrm{mi} / \mathrm{h}$ along a line north of east such that it remains due north of the first jet. After a half-hour, how fast are the jets separating?

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Problem 38

Solve the problems in related rates.
A car passes over a bridge at $15.0 \mathrm{m} / \mathrm{s}$ at the same time a boat passes under the bridge at a point $10.5 \mathrm{m}$ directly below the car. If the boat is moving perpendicularly to the bridge at $4.0 \mathrm{m} / \mathrm{s}$, how fast are the car and the boat separating 5.0 s later?

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Problem 39

Solve the problems in related rates.
A rope attached to a boat is being pulled in at a rate of $10.0 \mathrm{ft} / \mathrm{s}$. If the water is $20.0 \mathrm{ft}$ below the level at which the rope is being drawn in, how fast is the boat approaching the wharf when $36.0 \mathrm{ft}$ of rope are yet to be pulled in? See Fig. 24.26.

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Problem 40

Solve the problems in related rates.
A weather balloon leaves the ground 350 ft from an observer and rises vertically at $12 \mathrm{ft} / \mathrm{s}$. How fast is the line of sight from the observer to the balloon increasing when the balloon is $250 \mathrm{ft}$ high? See Fig. 24.27.

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Problem 41

Solve the problems in related rates.
A man $6.00 \mathrm{ft}$ tall approaches a street light $15.0 \mathrm{ft}$ above the ground at the rate of $5.00 \mathrm{ft} / \mathrm{s}$. How fast is the end of the man's shadow moving when he is $10.0 \mathrm{ft}$ from the base of the light? See Fig. 24.28 .

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Problem 42

Solve the problems in related rates.
A roller mechanism, as shown in Fig. $24.29,$ moves such that the right roller is always in contact with the bottom surface and the left roller is always in contact with the left surface. If the right roller is moving to the right at $1.50 \mathrm{cm} / \mathrm{s}$ when $x=10.0 \mathrm{cm},$ how fast is the left roller moving? (Hint: Use the law of cosines.)

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