Calculus AB
Section ________
Name ____________________________
Date _____________________________
~WORKSHEET 2-6 ON RELATED RATES
1. A paper cup, which is in the shape of a right circular cone, is 16 cm deep and has a radius of 4 cm. Water is
cm3
draining out of the cup at a constant rate of 2
. When the radius has a length of 3 cm, what is the rate of
sec
1


change of the radius?  Volume of a cone   r 2 h 
3


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in 3
2. A snowball is in the shape of a sphere. Its volume is increasing at a constant rate of 10
. How fast is the
min
4


radius increasing when the volume of the sphere is 36 in 3 ?  Volume of a sphere   r 3 
3


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3. All edges of a cube are expanding at a rate of 3 cm/sec. How fast is the volume of the cube changing when each
side is 10 centimeters?
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4. (1988) The figure shown represents an observer at point A watching balloon B
as it rises from point C. The balloon is rising at a constant rate of 3 m/sec, and
the observer is 100 m from point C.
(a) Find the rate of change in x at the instant when y = 50.
(b) Find the rate of change in the area of right triangle BCA at the instant
when y = 50.
(c) Find the rate of change of  at the instant when y = 50.
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5. A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the
scenario can be modeled with right triangles. At what rate is the length of the person's shadow
changing when the person is 16 ft from the lamppost?
Multiple Choice. All work should be shown.
6.
The sides of the rectangle above increase in such a way that
dz
 1 and
dx
3
dy
.
dt
dt
dt
dx
At the instant when x = 4 and y = 3, what is the value of
?
dt
1
(A)
(B) 1
(C) 2
(D) 5
(E) 5
3
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7. (1997) If  increases at a constant rate of 3 rad/min, at what rate is x increasing
in units/min when x = 3 units?
(A) 3
(B)
15
(C) 4
4
(D) 9
(E) 12
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8. (1998) If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at
a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?
(A) A is always increasing.
(C) A is decreasing only when b  h.
(E) A remains constant.
(B) A is always decreasing.
(D) A is decreasing when b  h.
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Answers
1 cm
1. 
18 sec
2.
5
in
18 min
dv
cm 3
 900
3.
dt
sec
4. (a)
5.
6. B
7. E
8. D
3 m
5 sec
dy 35

ft / sec
dt 13
(b) 150
m2
sec
(c)
3
rad
125 sec