Related Rates
Related Rate Equations
Suppose that a particle P͑x, y͒ is moving along a curve C in the plane so that its coordinates
x and y are differentiable functions of time t. If D is the distance from the origin to P,
then using the Chain Rule we can find an equation that relates dDրdt, dxրdt, and dyրdt.
D ϭ ͙xෆ2ෆϩෆ yෆ2ෆ
ᎏ
d
d
D
t
ᎏ ϭ ᎏ
1
2
ᎏ͑x2 ϩ y2͒Ϫ1ր2
(2xᎏ
d
d
x
t
ᎏ ϩ 2yᎏ
d
d
y
t
ᎏ)
Any equation involving two or more variables that are differentiable functions of time t
can be used to find an equation that relates their corresponding rates.
EXAMPLE 1 Finding Related Rate Equations
(a) Assume that the radius r of a sphere is a differentiable function of t and let V be the
volume of the sphere. Find an equation that relates dV/dt and dr/dt.
(b) Assume that the radius r and height h of a cone are differentiable functions of t and
let V be the volume of the cone. Find an equation that relates dVրdt, drրdt, and dhրdt.
SOLUTION
(a) V ϭ ᎏ
4
3
ᎏ pr3 Volume formula for a sphere
ᎏ
d
d
V
t
ᎏ ϭ 4pr2 ᎏ
d
d
r
t
ᎏ
(b) V ϭ ᎏ
p
3
ᎏr2h Cone volume formula
ᎏ
d
d
V
t
ᎏ ϭ ᎏ
p
3
ᎏ(r2 • ᎏ
d
d
h
t
ᎏ ϩ 2rᎏ
d
d
r
t
ᎏ • h)ϭ ᎏ
p
3
ᎏ(r2ᎏ
d
d
h
t
ᎏ ϩ 2rhᎏ
d
d
r
t
ᎏ)
Now try Exercise 3.
Solution Strategy
What has always distinguished calculus from algebra is its ability to deal with variables that
change over time. Example 1 illustrates how easy it is to move from a formula relating static
variables to a formula that relates their rates of change: simply differentiate the formula implicitly
with respect to t. This introduces an important category of problems called related
rate problems that still constitutes one of the most important applications of calculus.
We introduce a strategy for solving related rate problems, similar to the strategy we introduced
for max-min problems earlier in this chapter.
246 Chapter 4 Applications of Derivatives
4.6
What you’ll learn about
• Related Rate Equations
• Solution Strategy
• Simulating Related Motion
. . . and why
Related rate problems are at the
heart of Newtonian mechanics; it
was essentially to solve such
problems that calculus was
invented.
Strategy for Solving Related Rate Problems
1. Understand the problem. In particular, identify the variable whose rate of
change you seek and the variable (or variables) whose rate of change you know.
2. Develop a mathematical model of the problem. Draw a picture (many of these
problems involve geometric figures) and label the parts that are important to the
problem. Be sure to distinguish constant quantities from variables that change
over time. Only constant quantities can be assigned numerical values at the start.
continued
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 246
Section 4.6 Related Rates 247
We illustrate the strategy in Example 2.
EXAMPLE 2 A Rising Balloon
A hot-air balloon rising straight up from a level field is tracked by a range finder 500
feet from the lift-off point. At the moment the range finder’s elevation angle is p/4, the
angle is increasing at the rate of 0.14 radians per minute. How fast is the balloon rising
at that moment?
SOLUTION
We will carefully identify the six steps of the strategy in this first example.
Step 1: Let h be the height of the balloon and let u be the elevation angle.
We seek: dh/dt
We know: du/dt ϭ 0.14 rad/min
Step 2: We draw a picture (Figure 4.55). We label the horizontal distance “500 ft” because
it does not change over time. We label the height “h” and the angle of elevation
“u.” Notice that we do not label the angle “p/4,” as that would freeze
the picture.
Step 3: We need a formula that relates h and u. Since ᎏ
5
h
00
ᎏ ϭ tan u, we get
h ϭ 500 tan u.
Step 4: Differentiate implicitly:
ᎏ
d
d
t
ᎏ(h) ϭ ᎏ
d
d
t
ᎏ(500 tan u)
ᎏ
d
d
h
t
ᎏ ϭ 500 sec2 u ᎏ
d
d
u
t
ᎏ
Step 5: Let du/dt ϭ 0.14 and let u ϭ p/4. (Note that it is now safe to specify our moment
in time.)
ᎏ
d
d
h
t
ᎏ ϭ 500 sec2
΂ᎏ
p
4
ᎏ
΃(0.14) ϭ 500(͙2ෆ)2 (0.14) ϭ 140.
Step 6: At the moment in question, the balloon is rising at the rate of 140 ft/min.
Now try Exercise 11.
EXAMPLE 3 A Highway Chase
A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding
car that has turned the corner and is now moving straight east. When the cruiser is 0.6
mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar
that the distance between them and the car is increasing at 20 mph. If the cruiser is moving
at 60 mph at the instant of measurement, what is the speed of the car?
continued
3. Write an equation relating the variable whose rate of change you seek with
the variable(s) whose rate of change you know. The formula is often geometric,
but it could come from a scientific application.
4. Differentiate both sides of the equation implicitly with respect to time t. Be
sure to follow all the differentiation rules. The Chain Rule will be especially critical,
as you will be differentiating with respect to the parameter t.
5. Substitute values for any quantities that depend on time. Notice that it is only
safe to do this after the differentiation step. Substituting too soon “freezes the picture”
and makes changeable variables behave like constants, with zero derivatives.
6. Interpret the solution. Translate your mathematical result into the problem setting
(with appropriate units) and decide whether the result makes sense.
Figure 4.55 The picture shows how h
and u are related geometrically. We seek
dh/dt when u ϭ p/4 and du/dt ϭ 0.14
rad/min. (Example 2)
h
500 ft
Range finder
Balloon
θ
Unit Analysis in Example 2
A careful analysis of the units in Example
2 gives
dh/dt ϭ (500 ft)(͙2ෆ)2 (0.14 rad/min)
ϭ 140 ft и rad/min.
Remember that radian measure is actually
dimensionless, adaptable to whatever
unit is applied to the “unit” circle.
The linear units in Example 2 are measured
in feet, so “ft и rad ” is simply “ft.”
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 247
SOLUTION
We carry out the steps of the strategy.
Let x be the distance of the speeding car from the intersection, let y be the distance
of the police cruiser from the intersection, and let z be the distance between the car
and the cruiser. Distances x and z are increasing, but distance y is decreasing; so
dy/dt is negative.
We seek: dx/dt
We know: dz/dt ϭ 20 mph and dy/dt ϭ Ϫ60 mph
A sketch (Figure 4.56) shows that x, y, and z form three sides of a right triangle. We
need to relate those three variables, so we use the Pythagorean Theorem:
x2 ϩ y2 ϭ z2
Differentiating implicitly with respect to t, we get
2x ᎏ
d
d
x
t
ᎏ ϩ 2y ᎏ
d
d
y
t
ᎏ ϭ 2z ᎏ
d
d
z
t
ᎏ, which reduces to x ᎏ
d
d
x
t
ᎏ ϩ yᎏ
d
d
y
t
ᎏ ϭ zᎏ
d
d
z
t
ᎏ.
We now substitute the numerical values for x, y, dz/dt, dy/dt, and z (which equals ͙x2 ϩ y2ෆ ):
(0.8)ᎏ
d
d
x
t
ᎏ ϩ (0.6)(Ϫ60) ϭ ͙(0.8)2ෆϩ (0.6ෆ)2ෆ(20)
(0.8)ᎏ
d
d
x
t
ᎏ Ϫ 36 ϭ (1)(20)
ᎏ
d
d
x
t
ᎏ ϭ 70
At the moment in question, the car’s speed is 70 mph. Now try Exercise 13.
EXAMPLE 4 Filling a Conical Tank
Water runs into a conical tank at the rate of 9 ft3
րmin. The tank stands point down and
has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the
water is 6 ft deep?
SOLUTION 1
We carry out the steps of the strategy. Figure 4.57 shows a partially filled conical tank.
The tank itself does not change over time; what we are interested in is the changing
cone of water inside the tank. Let V be the volume, r the radius, and h the height of the
cone of water.
We seek: dh/dt
We know: dV/dt = 9 ft3/min
We need to relate V and h. The volume of the cone of water is V ϭ ᎏ
1
3
ᎏ pr2h, but this
formula also involves the variable r, whose rate of change is not given. We need to
either find dr/dt (see Solution 2) or eliminate r from the equation, which we can do by
using the similar triangles in Figure 4.57 to relate r and h:
ᎏ
h
r
ᎏ ϭ ᎏ
1
5
0
ᎏ, or simply r ϭ ᎏ
h
2
ᎏ.
Therefore,
V ϭ ᎏ
1
3
ᎏp
΂ᎏ
h
2
ᎏ
΃
2
h ϭ ᎏ
1
p
2
ᎏ h3.
continued
248 Chapter 4 Applications of Derivatives
Figure 4.56 A sketch showing the variables
in Example 3. We know dy/dt and
dz/dt, and we seek dx/dt. The variables x,
y, and z are related by the Pythagorean
Theorem: x2 ϩ y2 ϭ z2.
y
x
z
Figure 4.57 In Example 4, the cone of
water is increasing in volume inside the
reservoir. We know dV/dt and we seek
dh/dt. Similar triangles enable us to relate
V directly to h.
h 10 ft
5 ft
r
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 248
Section 4.6 Related Rates 249
Differentiate with respect to t:
ᎏ
d
d
V
t
ᎏ ϭ ᎏ
1
p
2
ᎏ • 3h2 ᎏ
d
d
h
t
ᎏ ϭ ᎏ
p
4
ᎏ h2 ᎏ
d
d
h
t
ᎏ.
Let h ϭ 6 and dV/dt ϭ 9; then solve for dh/dt:
9 ϭ ᎏ
p
4
ᎏ(6)2 ᎏ
d
d
h
t
ᎏ
ᎏ
d
d
h
t
ᎏ ϭ ᎏ
p
1
ᎏ Ϸ 0.32
At the moment in question, the water level is rising at 0.32 ft/min.
SOLUTION 2
The similar triangle relationship
r ϭ ᎏ
h
2
ᎏ also implies that ᎏ
d
d
r
t
ᎏ ϭ ᎏ
1
2
ᎏ ᎏ
d
d
h
t
ᎏ
and that r ϭ 3 when h ϭ 6. So, we could have left all three variables in the formula
V ϭ ᎏ
1
3
ᎏ␲r2h and proceeded as follows:
ᎏ
d
d
V
t
ᎏ ϭ ᎏ
1
3
ᎏp
΂2r ᎏ
d
d
r
t
ᎏhϩr2 ᎏ
d
d
h
t
ᎏ
΃
ϭ ᎏ
1
3
ᎏp
΂2r
΂ᎏ
1
2
ᎏ ᎏ
d
d
h
t
ᎏ
΃hϩr2 ᎏ
d
d
h
t
ᎏ
΃
9 ϭ ᎏ
1
3
ᎏp
΂2(3)
΂ᎏ
1
2
ᎏ ᎏ
d
d
h
t
ᎏ
΃(6)ϩ(3)2 ᎏ
d
d
h
t
ᎏ
΃
9 ϭ 9p ᎏ
d
d
h
t
ᎏ
ᎏ
d
d
h
t
ᎏ ϭ ᎏ
p
1
ᎏ
This is obviously more complicated than the one-variable approach. In general, it is computationally
easier to simplify expressions as much as possible before you differentiate.
Now try Exercise 17.
Simulating Related Motion
Parametric mode on a grapher can be used to simulate the motion of moving objects when
the motion of each can be expressed as a function of time. In a classic related rate problem,
the top end of a ladder slides vertically down a wall as the bottom end is pulled horizontally
away from the wall at a steady rate. Exploration 1 shows how you can use your
grapher to simulate the related movements of the two ends of the ladder.
The Sliding Ladder
A 10-foot ladder leans against a vertical wall. The base of the ladder is pulled away
from the wall at a constant rate of 2 ft/sec.
1. Explain why the motion of the two ends of the ladder can be represented by the
parametric equations given on the next page. continued
EXPLORATION 1
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 249
250 Chapter 4 Applications of Derivatives
Figure 4.58 This 5-step program (with the viewing window set as shown) will animate the ladder
in Exploration 1. Be sure any functions in the “Yϭ” register are turned off. Run the program and the
ladder appears against the wall; push ENTER to start the bottom moving away from the wall.
For an enhanced picture, you can insert the commands “:Pt-On(2,2ϩ͙ෆ(100Ϫ(2A)2),2)”
and “:Pt-On(2ϩ2A,2,2)” on either side of the middle line of the program.
PROGRAM
For (A, 0, 5, .25)
ClrDraw
Line(2,2+ (100–
(2A)2), 2+2A, 2)
If A=0 Pause
End
: LADDER
:
:
:
:
:
√
WINDOW
Xmin=2
Xmax=20
Xscl=0
Ymin=1
Ymax=13
Yscl=0
Xres=1
:
X1T ϭ 2T
Y1T ϭ 0
X2T ϭ 0
Y2T ϭ ͙102 Ϫෆ(2T)2ෆ
2. What minimum and maximum values of T make sense in this problem?
3. Put your grapher in parametric and simultaneous modes. Enter the parametric
equations and change the graphing style to “0” (the little ball) if your grapher
has this feature. Set Tminϭ0, Tmaxϭ5, Tstepϭ5/20, XminϭϪ1,
Xmaxϭ17, Xsclϭ0, YminϭϪ1, Ymaxϭ11, and Ysclϭ0. You can speed up
the action by making the denominator in the Tstep smaller or slow it down by
making it larger.
4. Press GRAPH and watch the two ends of the ladder move as time changes. Do
both ends seem to move at a constant rate?
5. To see the simulation again, enter “ClrDraw” from the DRAW menu.
6. If y represents the vertical height of the top of the ladder and x the distance of
the bottom from the wall, relate y and x and find dyրdt in terms of x and y. (Remember
that dxրdt ϭ 2.)
7. Find dyրdt when t ϭ 3 and interpret its meaning. Why is it negative?
8. In theory, how fast is the top of the ladder moving as it hits the ground?
Figure 4.58 shows you how to write a calculator program that animates the falling ladder
as a line segment.
Quick Review 4.6 (For help, go to Sections 1.1, 1.4, and 3.7.)
In Exercises 1 and 2, find the distance between the points
A and B.
1. A͑0, 5͒, B͑7, 0͒ ͙74ෆ 2. A͑0, a͒, B͑b, 0͒ ͙a2 ϩ bෆ2ෆ
In Exercises 3–6, find dyրdx.
3. 2xy ϩ y2 ϭ x ϩ y ᎏ
2x
1
ϩ
Ϫ
2y
2
Ϫ
y
1
ᎏ
4. x sin y ϭ 1 Ϫ xy Ϫᎏ
x
y
ϩ
ϩ
x
s
c
in
os
y
y
ᎏ
5. x2 ϭ tan y 2x cos2y 6. ln ͑x ϩ y͒ ϭ 2x 2x ϩ 2y Ϫ 1
In Exercises 7 and 8, find a parametrization for the line segment with
endpoints A and B.
7. A͑Ϫ2, 1͒, B͑4, Ϫ3͒ 8. A͑0, Ϫ4͒, B͑5, 0͒
In Exercises 9 and 10, let x ϭ 2 cos t, y ϭ 2 sin t. Find a parameter
interval that produces the indicated portion of the graph.
9. The portion in the second and third quadrants, including the
points on the axes.
10. The portion in the fourth quadrant, including the points on the axes.
8. One possible answer: x ϭ 5t, y ϭ Ϫ4 ϩ 4t, 0 Յ t Յ 1.
One possible answer: p/2 Յ t Յ 3p/2
One possible answer: 3p/2 Յ t Յ2p
7. One possible answer: x ϭ Ϫ 2 ϩ 6t, y ϭ 1 Ϫ 4t, 0 Յ t Յ 1.
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 250
Section 4.6 Related Rates 251
Section 4.6 Exercises
In Exercises 1–41, assume all variables are differentiable functions
of t.
1. Area The radius r and area A of a circle are related by the
equation A ϭ pr2. Write an equation that relates dAրdt
to drրdt. ᎏ
d
d
A
t
ᎏ ϭ 2pr ᎏ
d
d
r
t
ᎏ
2. Surface Area The radius r and surface area S of a sphere are
related by the equation S ϭ 4pr2. Write an equation that
relates dSրdt to drրdt. ᎏ
d
d
S
t
ᎏ ϭ 8prᎏ
d
d
r
t
ᎏ
3. Volume The radius r, height h, and volume V of a right
circular cylinder are related by the equation V ϭ pr2h.
(a) How is dVրdt related to dhրdt if r is constant?
(b) How is dVրdt related to drրdt if h is constant?
(c) How is dVրdt related to drրdt and dhրdt if neither r nor h
is constant?
4. Electrical Power The power P (watts) of an electric circuit is
related to the circuit’s resistance R (ohms) and current
I (amperes) by the equation P ϭ RI2.
(a) How is dPրdt related to dRրdt and dIրdt?
(b) How is dRրdt related to dIրdt if P is constant?
5. Diagonals If x, y, and z are lengths of the edges of a
rectangular box, the common length of the box’s diagonals is
s ϭ ͙x2 ϩ yෆ2 ϩ z2ෆ. How is dsրdt related to dxրdt, dyրdt, and
dzրdt? See below.
6. Area If a and b are the lengths of two sides of a triangle, and u
the measure of the included angle, the area A of the triangle is
A ϭ ͑1ր2͒ ab sin u. How is dAրdt related to daրdt, dbրdt, and
duրdt?
7. Changing Voltage The voltage V (volts), current I (amperes),
and resistance R (ohms) of an electric circuit like the one shown
here are related by the equation V ϭ IR. Suppose that V is
increasing at the rate of 1 voltրsec while I is decreasing at the
rate of 1ր3 ampրsec. Let t denote time in sec.
(a) What is the value of dVրdt?
(b) What is the value of dIրdt?
(c) Write an equation that relates dRրdt to dVրdt and dIրdt.
(d) Writing to Learn Find the rate at which R is changing
when V ϭ 12 volts and I ϭ 2 amp. Is R increasing, or
decreasing? Explain.
8. Heating a Plate When a circular plate of metal is heated
in an oven, its radius increases at the rate of 0.01 cmրsec.
At what rate is the plate’s area increasing when the radius
is 50 cm? p cm2/sec
9. Changing Dimensions in a Rectangle The length ᐍ of a
rectangle is decreasing at the rate of 2 cmրsec while the width w
is increasing at the rate of 2 cmրsec. When ᐍ ϭ 12 cm and
w ϭ 5 cm, find the rates of change of See page 255.
(a) the area, (b) the perimeter, and
(c) the length of a diagonal of the rectangle.
(d) Writing to Learn Which of these quantities are
decreasing, and which are increasing? Explain.
10. Changing Dimensions in a Rectangular Box Suppose
that the edge lengths x, y, and z of a closed rectangular box are
changing at the following rates:
ᎏ
d
d
x
t
ᎏ ϭ 1 mրsec, ᎏ
d
d
y
t
ᎏ ϭ Ϫ2 mրsec, ᎏ
d
d
z
t
ᎏ ϭ 1 mրsec.
Find the rates at which the box’s (a) volume, (b) surface area,
and (c) diagonal length s ϭ ͙xෆ2ෆϩෆ yෆ2ෆϩෆ zෆ2ෆ are changing at the
instant when x ϭ 4, y ϭ 3, and z ϭ 2.
11. Inflating Balloon A spherical balloon is inflated with helium
at the rate of 100p ft3
րmin.
(a) How fast is the balloon’s radius increasing at the instant the
radius is 5 ft? 1 ft/min
(b) How fast is the surface area increasing at that instant?
12. Growing Raindrop Suppose that a droplet of mist is a perfect
sphere and that, through condensation, the droplet picks up
moisture at a rate proportional to its surface area. Show that
under these circumstances the droplet’s radius increases at a
constant rate. See page 255.
13. Air Traffic Control An airplane is flying at an altitude of
7 mi and passes directly over a radar antenna as shown
in the figure. When the plane is 10 mi from the antenna
͑s ϭ 10͒, the radar detects that the distance s is changing at the
rate of 300 mph. What is the speed of the airplane at
that moment? ᎏ
d
d
x
t
ᎏ ϭ mph Ϸ 420.08 mph
14. Flying a Kite Inge flies a kite at a height of 300 ft, the wind
carrying the kite horizontally away at a rate of 25 ftրsec. How
fast must she let out the string when the kite is 500 ft away from
her? 20 ft/sec
15. Boring a Cylinder The mechanics at Lincoln Automotive are
reboring a 6-in. deep cylinder to fit a new piston. The machine
they are using increases the cylinder’s radius one-thousandth of
an inch every 3 min. How rapidly is the cylinder volume
increasing when the bore (diameter) is 3.800 in.?
3000
ᎏ
͙51ෆ
V+ –
R
I
y
x
x
s7 mi
ᎏ
d
d
V
t
ᎏ ϭ pr2 ᎏ
d
d
h
t
ᎏ
ᎏ
d
d
V
t
ᎏ ϭ 2prhᎏ
d
d
r
t
ᎏ
ᎏ
d
d
V
t
ᎏ ϭ pr2 ᎏ
d
d
h
t
ᎏ ϩ2prhᎏ
d
d
r
t
ᎏ
4. (a) ᎏ
d
d
P
t
ᎏ ϭ 2RI ᎏ
d
d
I
t
ᎏ ϩ I2 ᎏ
d
d
R
t
ᎏ (b) 0 ϭ 2RI ᎏ
d
d
I
t
ᎏ ϩ I2 ᎏ
d
d
R
t
ᎏ, or, ᎏ
d
d
R
t
ᎏ ϭ
΂Ϫᎏ
2
I
R
ᎏ
΃΂ᎏ
d
d
I
t
ᎏ
΃ϭ
΂Ϫᎏ
2
I
P
3
ᎏ
΃΂ᎏ
d
d
I
t
ᎏ
΃
ᎏ
d
d
A
t
ᎏ ϭ ᎏ
1
2
ᎏ
΂b sin ␪ ᎏ
d
d
a
t
ᎏ ϩ a sin ␪ ᎏ
d
d
b
t
ᎏ ϩ ab cos ␪ ᎏ
d
d
␪
t
ᎏ
΃
(a) 1 volt/sec
(b) Ϫᎏ
1
3
ᎏ amp/sec
(c) ᎏ
d
d
V
t
ᎏ ϭ Iᎏ
d
d
R
t
ᎏ ϩ Rᎏ
d
d
I
t
ᎏ
(d) ᎏ
d
d
R
t
ᎏ ϭ ᎏ
3
2
ᎏ ohms/sec. R is
increasing since ᎏ
d
d
R
t
ᎏ is positive.
(a) 2 m3/sec (b) 0 m2/sec
(c) 0 m/sec
40p ft2/min
ᎏ
2
1
5
9
0
p
0
ᎏ Ϸ 0.0239 in3/min
5. ᎏ
d
d
s
t
ᎏ ϭ
x ᎏ
d
d
x
t
ᎏ ϩ y ᎏ
d
d
y
t
ᎏ ϩ z ᎏ
d
d
z
t
ᎏ
ᎏᎏᎏ
͙x2 ϩ yෆ2 ϩ z2ෆ
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252 Chapter 4 Applications of Derivatives
16. Growing Sand Pile Sand falls from a conveyor belt at the
rate of 10 m3
րmin onto the top of a conical pile. The height of
the pile is always three-eighths of the base diameter. How fast
are the (a) height and (b) radius changing when the pile is 4 m
high? Give your answer in cmրmin.
17. Draining Conical Reservoir Water is flowing at the rate of
50 m3
րmin from a concrete conical reservoir (vertex down) of
base radius 45 m and height 6 m. (a) How fast is the water level
falling when the water is 5 m deep? (b) How fast is the radius of
the water’s surface changing at that moment? Give your answer
in cmրmin.
18. Draining Hemispherical Reservoir Water is flowing at the
rate of 6 m3
րmin from a reservoir shaped like a hemispherical bowl
of radius 13 m, shown here in profile. Answer the following
questions given that the volume of water in a hemispherical bowl of
radius R is V ϭ ͑pր3͒y2͑3R Ϫ y͒ when the water is y units deep.
(a) At what rate is the water level changing when the water is
8 m deep?
(b) What is the radius r of the water’s surface when the water is
y m deep? r ϭ ͙169 Ϫෆ (13 Ϫෆ y)2ෆ ϭ ͙26y Ϫෆ y2ෆ
(c) At what rate is the radius r changing when the water is
8 m deep?
19. Sliding Ladder A 13-ft ladder is leaning against a house (see
figure) when its base starts to slide away. By the time the base is
12 ft from the house, the base is moving at the rate of 5 ftրsec.
20. Filling a Trough A trough is 15 ft long and 4 ft across the top
as shown in the figure. Its ends are isosceles triangles with
height 3 ft. Water runs into the trough at the rate of 2.5 ft3
րmin.
How fast is the water level rising when it is 2 ft deep? ᎏ
1
1
6
ᎏ ft/min
21. Hauling in a Dinghy A dinghy is pulled toward a dock by a
rope from the bow through a ring on the dock 6 ft above the bow
as shown in the figure. The rope is hauled in at the rate of 2
ftրsec.
(a) How fast is the boat approaching
the dock when 10 ft of rope are out? ᎏ
5
2
ᎏ ft/sec
(b) At what rate is angle ␪
changing at that moment?
22. Rising Balloon A balloon is rising vertically above a level,
straight road at a constant rate of 1 ftրsec. Just when the balloon
is 65 ft above the ground, a bicycle moving at a constant rate of
17 ftրsec passes under it. How fast is the distance between the
bicycle and balloon increasing 3 sec later (see figure)? 11 ft/sec
In Exercises 23 and 24, a particle is moving along the curve
y ϭ f ͑x͒.
23. Let y ϭ f ͑x͒ ϭ ᎏ
1 ϩ
10
x2ᎏ.
If dxրdt ϭ 3 cmրsec, find dyրdt at the point where
(a) x ϭ Ϫ2. (b) x ϭ 0. (c) x ϭ 20.
24. Let y ϭ f ͑x͒ ϭ x3 Ϫ 4x.
If dxրdt ϭ Ϫ2 cmրsec, find dyրdt at the point where
(a) x ϭ Ϫ3. (b) x ϭ 1. (c) x ϭ 4.
r
y
13 m
Water level
Center of sphere
x
x(t)0
u
y(t)
y
13-ft ladder
2 ft
3 ft
15 ft
2.5 ft3/min
4 ft
Ring at edge
of dock
6
u
'
x(t)
s(t)
y(t)
y
x
0
(a) How fast is the top of the ladder sliding down the wall
at that moment? 12 ft/sec
(b) At what rate is the area of the triangle formed by the ladder,
wall, and ground changing at that moment? Ϫᎏ
11
2
9
ᎏ ft2/sec
(c) At what rate is the angle u between the ladder and the ground
changing at that moment? Ϫ1 radian/sec
(a) ᎏ
1
3
1
2
2
p
5
ᎏ Ϸ 11.19 cm/min (b) ᎏ
3
8
7
p
5
ᎏ Ϸ 14.92 cm/min
(a) ᎏ
9
3
p
2
ᎏ Ϸ 1.13 cm/min (b) Ϫᎏ
3
8
p
0
ᎏ Ϸ Ϫ8.49 cm/min
Ϫᎏ
24
1
p
ᎏ Ϸ Ϫ0.01326 m/min or Ϫᎏ
6
2
p
5
ᎏ Ϸ Ϫ1.326 cm/min
Ϫᎏ
28
5
8p
ᎏ Ϸ Ϫ0.00553 m/min or Ϫᎏ
7
1
2
2
p
5
ᎏ Ϸ Ϫ0.553 cm/min
Ϫᎏ
2
3
0
ᎏ radian/sec
(a) Ϫ46 cm/sec (b) 2 cm/sec (c) Ϫ88 cm/sec
(a) ᎏ
2
5
4
ᎏ cm/sec (b) 0 cm/sec
(c) Ϫᎏ
16
1
0
2
,
0
8
0
01
ᎏ Ϸ Ϫ0.00746 cm/sec
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 252
Section 4.6 Related Rates 253
25. Particle Motion A particle moves along the parabola y ϭ x2
in the first quadrant in such a way that its x-coordinate (in
meters) increases at a constant rate of 10 mրsec. How fast is the
angle of inclination u of theline joining the particle to the origin
changing when x ϭ 3? 1 radian/sec
26. Particle Motion A particle moves from right to left
along the parabolic curve y ϭ ͙Ϫෆxෆ in such a way that its
x-coordinate (in meters) decreases at the rate of 8 mրsec. How
fast is the angle of inclination u of the line joining the particle to
the origin changing when x ϭ Ϫ4? ᎏ
2
5
ᎏ radian/sec
27. Melting Ice A spherical iron ball is coated with a layer
of ice of uniform thickness. If the ice melts at the rate of
8 mLրmin, how fast is the outer surface area of ice decreasing
when the outer diameter (ball plus ice) is 20 cm? 1.6 cm2/min
28. Particle Motion A particle P͑x, y͒ is moving in the coordinate
plane in such a way that dxրdt ϭ Ϫ1 mրsec and
dyրdt ϭ Ϫ5 mրsec. How fast is the particle’s distance from
the origin changing as it passes through the point ͑5, 12͒?
29. Moving Shadow A man 6 ft tall walks at the rate of
5 ftրsec toward a streetlight that is 16 ft above the ground.
At what rate is the length of his shadow changing when he
is 10 ft from the base of the light? Ϫ3 ft/sec
30. Moving Shadow A light shines from the top of a pole
50 ft high. A ball is dropped from the same height from a
point 30 ft away from the light as shown below. How fast is
the ball’s shadow moving along the ground 1ր2 sec later?
(Assume the ball falls a distance s ϭ 16t2 in t sec.) Ϫ1500 ft/sec
31. Moving Race Car You are videotaping a race from a
stand 132 ft from the track, following a car that is moving
at 180 mph (264 ftրsec) as shown in the figure. About how fast
will your camera angle u be changing when the car is right in
front of you? a half second later?
32. Speed Trap A highway patrol airplane flies 3 mi above a level,
straight road at a constant rate of 120 mph. The pilot sees an
oncoming car and with radar determines that at the instant the
line-of-sight distance from plane to car is 5 mi the line-of-sight
distance is decreasing at the rate of 160 mph. Find the car’s speed
along the highway. 80 mph
33. Building’s Shadow On a morning of a day when the sun will
pass directly overhead, the shadow of an 80-ft building on level
ground is 60 ft long as shown in the figure. At the moment in
question, the angle u the sun makes with the ground is
increasing at the rate of 0.27°րmin. At what rate is the shadow
length decreasing? Express your answer in in.րmin, to the
nearest tenth. (Remember to use radians.) 7.1 in./min
34. Walkers A and B are walking on straight streets that meet
at right angles. A approaches the intersection at 2 mրsec and
B moves away from the intersection at 1 mրsec as shown in the
figure. At what rate is the angle u changing when A is 10 m
from the intersection and B is 20 m from the intersection?
Express your answer in degrees per second to the nearest
degree. Ϫ6 deg/sec
35. Moving Ships Two ships are steaming away from a point O
along routes that make a 120° angle. Ship A moves at
14 knots (nautical miles per hour; a nautical mile is 2000 yards).
Ship B moves at 21 knots. How fast are the ships moving apart
when OA ϭ 5 and OB ϭ 3 nautical miles? 29.5 knots
x
Light
30 x(t)
1/2 sec later
Shadow
Ball at time t ϭ 0
NOT TO SCALE
0
50-ft
pole
Race Car
132'
Camera
θ
NOT TO SCALE
Plane
x(t)
Car
0
x
3 mi
s(t)
80'
␪
60'
O
␪
A
B
Ϫ5 m/sec
In front: 2 radians/sec;
Half second later: 1 radian/sec
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 253
254 Chapter 4 Applications of Derivatives
Standardized Test Questions
You may use a graphing calculator to solve the following
problems.
36. True or False If the radius of a circle is expanding at a
constant rate, then its circumference is increasing at a constant
rate. Justify your answer.
37. True or False If the radius of a circle is expanding at a
constant rate, then its area is increasing at a constant rate. Justify
your answer.
38. Multiple Choice If the volume of a cube is increasing at 24
in3/ min and each edge of the cube is increasing at 2 in./min,
what is the length of each edge of the cube? A
(A) 2 in. (B) 2͙2ෆ in. (C) ͙
3
12ෆ in. (D) 4 in. (E) 8 in.
39. Multiple Choice If the volume of a cube is increasing at
24 in3/ min and the surface area of the cube is increasing at
12 in2/ min, what is the length of each edge of the cube? E
(A) 2 in. (B) 2͙2ෆ in. (C) ͙
3
12ෆ in. (D) 4 in. (E) 8 in.
40. Multiple Choice A particle is moving around the unit circle
(the circle of radius 1 centered at the origin). At the point (0.6,
0.8) the particle has horizontal velocity dx/dt ϭ 3. What is its
vertical velocity dy/dt at that point? C
(A) Ϫ3.875 (B) Ϫ3.75 (C) Ϫ2.25 (D) 3.75 (E) 3.875
41. Multiple Choice A cylindrical rubber cord is stretched at a
constant rate of 2 cm per second. Assuming its volume does not
change, how fast is its radius shrinking when its length is 100 cm
and its radius is 1 cm? B
(A) 0 cm/sec (B) 0.01 cm/sec (C) 0.02 cm/sec
(D) 2 cm/sec (E) 3.979 cm/sec
Explorations
42. Making Coffee Coffee is draining from a conical filter into a
cylindrical coffeepot at the rate of 10 in3
րmin.
43. Cost, Revenue, and Profit A company can manufacture
x items at a cost of c͑x͒ dollars, a sales revenue of r͑x͒ dollars,
and a profit of p͑x͒ ϭ r͑x͒ Ϫ c͑x͒ dollars (all amounts in
thousands). Find dcրdt, drրdt, and dpրdt for the following
values of x and dxրdt.
(a) r͑x͒ ϭ 9x, c͑x͒ ϭ x3 Ϫ 6x2 ϩ 15x,
and dxրdt ϭ 0.1 when x ϭ 2.
(b) r͑x͒ ϭ 70x, c͑x͒ ϭ x3 Ϫ 6x2 ϩ 45րx,
and dxրdt ϭ 0.05 when x ϭ 1.5.
44. Group Activity Cardiac Output In the late 1860s, Adolf
Fick, a professor of physiology in the Faculty of Medicine in
Würtzberg, Germany, developed one of the methods we use
today for measuring how much blood your heart pumps in a
minute. Your cardiac output as you read this sentence is
probably about 7 liters a minute. At rest it is likely to be a bit
under 6 Lրmin. If you are a trained marathon runner running a
marathon, your cardiac output can be as high as 30 Lրmin.
Your cardiac output can be calculated with the formula
y ϭ ᎏ
Q
D
ᎏ,
where Q is the number of milliliters of CO2 you exhale
in a minute and D is the difference between the CO2
concentration (mLրL) in the blood pumped to the lungs and the
CO2 concentration in the blood returning from the lungs. With
Q ϭ 233 mLրmin and D ϭ 97 Ϫ 56 ϭ 41 mLրL,
y ϭ Ϸ 5.68 Lրmin,
fairly close to the 6 Lրmin that most people have at basal
(resting) conditions. (Data courtesy of J. Kenneth Herd, M.D.,
Quillan College of Medicine, East Tennessee State University.)
Suppose that when Q ϭ 233 and D ϭ 41, we also know that
D is decreasing at the rate of 2 units a minute but that
Q remains unchanged. What is happening to the cardiac output?
Extending the Ideas
45. Motion along a Circle A wheel of radius 2 ft makes
8 revolutions about its center every second.
(a) Explain how the parametric equations
x ϭ 2 cos u, y ϭ 2 sin u
can be used to represent the motion of the wheel.
(b) Express u as a function of time t.
(c) Find the rate of horizontal movement and the rate of vertical
movement of a point on the edge of the wheel when it is at the
position given by u ϭ pր4, pր2, and p.
46. Ferris Wheel A Ferris wheel with radius 30 ft makes one
revolution every 10 sec.
(a) Assume that the center of the Ferris wheel is located at the
point ͑0, 40͒, and write parametric equations to model its
motion. [Hint: See Exercise 45.]
(b) At t ϭ 0 the point P on the Ferris wheel is located at ͑30, 40͒.
Find the rate of horizontal movement, and the rate of vertical
movement of the point P when t ϭ 5 sec and t ϭ 8 sec.
233 mLրmin
ᎏᎏ
41 mLրL
6"
6"
How fast
is this
level falling?
How fast
is this
level rising?
6"
(a) How fast is the level in the pot rising when the coffee in the
cone is 5 in. deep?
(b) How fast is the level in the cone falling at that moment?
36. True. Since ᎏ
d
d
C
t
ᎏ ϭ 2p ᎏ
d
d
r
t
ᎏ, a constant ᎏ
d
d
r
t
ᎏ results in a constant ᎏ
d
d
C
t
ᎏ. 37. False. Since ᎏ
d
d
A
t
ᎏ ϭ 2pr ᎏ
d
d
r
t
ᎏ, the value of ᎏ
d
d
A
t
ᎏ depends on r.
ᎏ
d
d
y
t
ᎏ ϭ ᎏ
1
4
6
6
8
6
1
ᎏ Ϸ 0.277 L/min2
ᎏ
d
d
c
t
ᎏ ϭ 0.3 ᎏ
d
d
r
t
ᎏ ϭ 0.9 ᎏ
d
d
p
t
ᎏ ϭ 0.6
ᎏ
d
d
c
t
ᎏ ϭ Ϫ1.5625
ᎏ
d
d
r
t
ᎏ ϭ 3.5 ᎏ
d
d
p
t
ᎏ ϭ 5.0625
(a) ᎏ
9
1
p
0
ᎏ Ϸ 0.354 in./min
(b) ᎏ
5
8
p
ᎏ Ϸ 0.509 in./min
5128_Ch04_pp186-260.qxd 01/16/06 11:59 AM Page 254
Section 4.6 Related Rates 255
47. Industrial Production (a) Economists often use the
expression “rate of growth” in relative rather than absolute
terms. For example, let u ϭ f ͑t͒ be the number of people in
the labor force at time t in a given industry. (We treat this
function as though it were differentiable even though it is an
integer-valued step function.) 9% per year
Let v ϭ g͑t͒ be the average production per person in the labor
force at time t. The total production is then y ϭ uv.
If the labor force is growing at the rate of 4% per year ͑duրdt ϭ
0.04u͒ and the production per worker is growing at the rate of
5% per year ͑dvրdt ϭ 0.05v͒, find the rate of growth of the total
production, y.
(b) Suppose that the labor force in part (a) is decreasing at the
rate of 2% per year while the production per person is increasing
at the rate of 3% per year. Is the total production increasing, or
is it decreasing, and at what rate? Increasing at 1% per year
Quick Quiz for AP* Preparation: Sections 4.4–4.6
You may use a graphing calculator to solve the following
problems.
1. Multiple Choice If Newton's method is used to approximate
the real root of x3 ϩ 2x Ϫ 1 ϭ 0, what would the third approximation,
x3, be if the first approximation is x1 ϭ 1? B
(A) 0.453 (B) 0.465 (C) 0.495 (D) 0.600 (E) 1.977
2. Multiple Choice The sides of a right triangle with legs x and
y and hypotenuse z increase in such a way that dzրdt ϭ 1 and
dxրdt ϭ 3 dyրdt. At the instant when x ϭ 4 and y ϭ 3, what is
dxրdt ? B
(A) ᎏ
1
3
ᎏ (B) 1 (C) 2 (D) ͙5ෆ (E) 5
3. Multiple Choice An observer 70 meters south of a railroad
crossing watches an eastbound train traveling at 60 meters per
second. At how many meters per second is the train moving
away from the observer 4 seconds after it passes through the intersection?
A
(A) 57.60 (B) 57.88 (C) 59.20 (D) 60.00 (E) 67.40
4. Free Response (a)Approximate ͙26ෆ by using the linearization
of y ϭ ͙xෆ at the point (25, 5). Show the computation that leads
to your conclusion.
(b) Approximate ͙26ෆ by using a first guess of 5 and one iteration
of Newton's method to approximate the zero of x2 Ϫ 26.
Show the computation that leads to your conclusion.
(c) Approximate ͙
3
26ෆ by using an appropriate linearization.
Show the computation that leads to your conclusion.
Chapter 4 Key Terms
absolute change (p. 240)
absolute maximum value (p. 187)
absolute minimum value (p. 187)
antiderivative (p. 200)
antidifferentiation (p. 200)
arithmetic mean (p. 204)
average cost (p. 224)
center of linear approximation (p. 233)
concave down (p. 207)
concave up (p. 207)
concavity test (p. 208)
critical point (p. 190)
decreasing function (p. 198)
differential (p. 237)
differential estimate of change (p. 239)
differential of a function (p. 239)
extrema (p. 187)
Extreme Value Theorem (p. 188)
first derivative test (p. 205)
first derivative test for local extrema (p. 205)
geometric mean (p. 204)
global maximum value (p. 177)
global minimum value (p. 177)
increasing function (p. 198)
linear approximation (p. 233)
linearization (p. 233)
local linearity (p. 233)
local maximum value (p. 189)
local minimum value (p. 189)
logistic curve (p. 210)
logistic regression (p. 211)
marginal analysis (p. 223)
marginal cost and revenue (p. 223)
Mean Value Theorem (p. 196)
monotonic function (p. 198)
Newton’s method (p. 235)
optimization (p. 219)
percentage change (p. 240)
point of inflection (p. 208)
profit (p. 223)
quadratic approximation (p. 245)
related rates (p. 246)
relative change (p. 240)
relative extrema (p. 189)
Rolle’s Theorem (p. 196)
second derivative test
for local extrema (p. 211)
standard linear approximation (p. 233)
9. (a) ᎏ
d
d
A
t
ᎏ ϭ 14 cm2/sec (b) ᎏ
d
d
P
t
ᎏ ϭ 0 cm/sec
(c) ᎏ
d
d
D
t
ᎏ ϭ Ϫᎏ
1
1
4
3
ᎏ cm/sec
(d) The area is increasing, because its derivative is positive.
The perimeter is not changing, because its derivative is zero.
The diagonal length is decreasing, because its derivative is negative.
12. V ϭ ᎏ
4
3
ᎏpr3, so ᎏ
d
d
V
t
ᎏ ϭ 4pr2 ᎏ
d
d
r
t
ᎏ. But S ϭ 4pr2, so we are given that
ᎏ
d
d
V
t
ᎏ ϭ kS ϭ 4kpr2. Substituting, 4kpr2 ϭ 4pr2 ᎏ
d
d
r
t
ᎏ which gives ᎏ
d
d
r
t
ᎏ ϭ k.
5128_Ch04_pp186-260.qxd 1/13/06 12:37 PM Page 255