Ch 4.4 Wkst AP Calc BC
Lesson
Related Rates
Steps:
1.
Draw a diagram and specify in mathematical form the rate of change you are looking for, and the specific conditions
(or values at a specific instant in time) that is involved in the problem.
 -3 and x  2.
Example: Find dt when dx
dt
Write an equation involving the variable whose rate of change is to be found.
dy
2.
 -3 and x  2 above.
Important: Do not use any value in the equation that is a specific condition, such as the dx
dt
d
The equation in Step 2 needs to have the same number of total variables as the number of dt ‘s. If not then . . .
a. find a secondary equation which consist of at least two variables, one being the “odd” variable. Then . . .
b. substitute the secondary equation into the equation in Step 2. Then . . .
c. simplify the equation.
Take the derivative implicitly of the equation in Step 3 with respect to time t .
Substitute all specific conditions and answer the question to the problem.
3.
4.
5.
Ex 1: A ladder 13 feet long is leaning against a wall. If the foot of the ladder is pulled away from the wall
at the rate of 0.5 feet per second, how fast will the top of the ladder be dropping at the instant when the
base is 5 feet from the wall?
(A)
1 ft/sec
 12
(B)
 81 ft/sec
(C)

(D)

(E)

1
ft/sec
6
5 ft/sec
24
1
ft/sec
4
13
y
x
*
Ex 2: Use the same situation in problem #41. Consider the area of the triangle created by the ladder and the
wall. How fast is this area changing at the instant when the base is 5 feet from the wall?
(A)
49
4
ft²/sec
(B)
27
6
ft²/sec
(C)
152
7
ft²/sec
(D)
193
64
ft²/sec
(E)
119
48
ft²/sec *
Ch 4.4 Wkst AP Calc BC
Name:
Related Rates
Steps:
1.
Draw a diagram and specify in mathematical form the rate of change you are looking for, and the specific conditions
(or values at a specific instant in time) that is involved in the problem.
 -3 and x  2.
Example: Find dt when dx
dt
Write an equation involving the variable whose rate of change is to be found.
dy
2.
 -3 and x  2 above.
Important: Do not use any value in the equation that is a specific condition, such as the dx
dt
3.
5.
5.
d
The equation in Step 2 needs to have the same number of total variables as the number of dt ‘s. If not then . . .
c. find a secondary equation which consist of at least two variables, one being the “odd” variable. Then . . .
d. substitute the secondary equation into the equation in Step 2. Then . . .
c. simplify the equation.
Take the derivative implicitly of the equation in Step 3 with respect to time t .
Substitute all specific conditions and answer the question to the problem.
1. A pebble thrown into a pond creates circular ripples such that the rate of change of the circumference is 12π cm/sec.
How fast is the area of the ripple changing when the radius is 3 cm?
Hint: A   r 2
(A) 6π cm²/sec
(B) 2π cm²/sec
C  2 r
(C) 12π cm²/sec
(D) 36π cm²/sec
(E) 6 cm²/sec
2. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase
in the area of the circle at the instant when the circumference of the circle is 20π meters?
(A)
(B)
(C)
(D)
(E)
0.04π m²/sec
0.4π m²/sec
4π m²/sec
20π m²/sec
100π m²/sec
Hint: A   r 2
C  2 r
r
3. The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the square
root of the volume. Which of the following is a differential equation that describes this relationship?
(A) V (t )  k t
(B) V (t )  k V
(C)
dV
k t
dt
(D)
dV
k

dt
V
(E)
dV
k V
dt
4. The foot of a 20 ft ladder is being pulled away from a wall at the rate of 1.5 ft/sec. At the instant when the foot
is 12 ft away from the wall, the angle the ladder makes with the floor is decreasing at the rate (in radians/sec)
of:
1
3
1
3
3
(A)
(B)
(C)
(D)
(E)
8
50
16
40
32
5.
A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of
the coffee in the pot, measured in inches. How fast is the depth of coffee rising if the volume V of coffee in the
pot is changing at the rate of  5 h cubic inches per second. (The volume V of a cylinder with radius r and
height h is V   r 2 h .)
(A) 
h
5
in/sec
(B) 
h
3
in/sec
(C) 
h
5h
in/sec
(D) 
h
3h
in/sec
5 in
h
(E)  5 h in/sec
1998 AB #90D
6.
Peterson 1 #21D
7.
8. A water trough with vertical cross section in the shape of an equilateral triangle is being filled at a rate of
4 cubic feet per minute. Given that the trough is 12 feet long, how fast is the level of the water rising at
the instant that the water reaches a depth of 1.5 feet?
(A)
3
6
ft/min
(B)
3
9
ft/min
(C)
6
2
ft/min
(D)
6
3
ft/min
(E)
6
6
ft/min
Hint: Half an equilateral triangle is a 30-60-90
triangle. And all 30-60-90 triangles are similar
and therefore proportional. A 30-60-90
triangle is below.
2
30º
12 ft
3
60º
1
10 cm
r
10 cm
h
9. A container has the shape of an open right circular cone, as shown in the figure above. The height of the
container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that
3 cm/hr.
its depth h is changing at the rate of  10
( Vcone  13  r 2 h )
(a) Find the volume V of water in the container when h = 5 cm. Indicate units of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm.
Indicate units of measure.
(c) Show that the rate of change of the volume of water in the container due to evaporation is directly
proportional to the exposed surface area of the water. What is the constant of proportionality?
Peterson 1 #31E
10.
11. Let f ( x)  3x 4 and g ( x)  e 3 x  4 . At what value of x does f and g have the same rate of change?
(A) 0.127
(B) 0.204
(E) There are no such values.
(C) 0.455
(D) 0.649
1993 BC #34E
12.
ANSWERS:
1) D
8) B
2) C
9a) 125 
12
3) E
4) E
b)  15 
8
c)  3
10
5) A
10) E
6) D
11) B
7) D
12) E