AP Calculus Name
Review Related Rates
1. Suppose the radius of a spherical balloon is shrinking at ½ cm per minute. How fast is the volume
decreasing when the radius is 4 cm?
2. Suppose a snowball remains spherical while it melts, with the radius shrinking at 1 inch per hour.
How fast is the volume of the snowball decreasing when the radius is 2 inches?
3. Suppose the volume of the snowball in problem 2 shrinks at the rate of 2/V (cubic inches per
hour), so that dV/dt = -2/V. How fast is the radius changing when the radius is ½ inch?
4. A spherical balloon is inflated at the rate of 3 cubic centimeters per minute. How fast is the
radius of the balloon increasing when the radius is 6 cm?
5. Suppose a spherical balloon grows in such a way that after t seconds, V = 4 t cm3. How fast is
the radius changing after 64 seconds?
6. A spherical balloon is losing air at the rate of 2 cm3/min. How fast is the radius of the balloon
shrinking when the radius is 8 cm?
7. Water leaking onto a floor creates a circular pool with an area that increases at the rate of
3 cm2/min. How fast is the radius of the pool increasing when the radius is 10 cm?
8. A point moves around the circle x2 + y2 = 9. When the point is at (- 3 , 6 ), its x coordinate is
increasing at the rate of 20 units per second. How fast is its y coordinate changing at that
9. A ladder 15 feet long leans against a vertical wall. Suppose that when the bottom of the ladder is
x feet from the wall, the bottom is being pushed toward the wall at the rate of ½ x ft/sec. How
fast is the top of the ladder rising at the moment the bottom is 5 feet from the wall?
10. A board is 5 feet long slides down a wall. At the instant the bottom end is 4 feet from the wall,
the other end is moving down the wall at the rate of 2 ft/sec. At that moment:
a) How fast is the bottom end sliding along the ground?
b) How fast is the area of the region between the board, ground, and wall changing?
11. A water trough is 12 feet long, and its cross section is an equilateral triangle with sides 2 feet
long. Water is pumped into the trough at a rate of 3 ft3/min. How fast is the water level rising
when the depth of the water is ½ foot?
12. A beacon on a lighthouse 1 mile from shore revolves at the rate of 10 rad/min. Assuming that
the shoreline is straight, calculate the speed at which the spotlight is sweeping across the
shoreline as it lights up the sand 2 miles from the lighthouse. (Hint: dx/dt is the speed of the
image of the spotlight moving across the shoreline , and d /dt = 10 )
13. Boyle’s Law states that if the temperature of a gas remains constant, then the pressure p and the
volume v of the gas satisfy the equation pv=c where c is a constant. If the volume is decreasing at
the rate of 10 cm3/sec, how fast is the pressure increasing when the pressure is 100 lbs/cm 2 and
the volume is 20 cm3?
1) DV/dt = -32 cm3/min
2) DV/dt = -16in3/hr
3) Dr/dt = -12/ in/h
4) Dr/dt = 1/48 cm/min
5) Dr/dt = s
64( π 3 )(3 3 )
6) Dr/dt =
7) Dr/dt =
8) Dy/dt = 10 2 u sec
5 2 ft
9) Dy/dt = sec
3 ft 2
10) a) b) dA 7 in sec
2 sec dt 4
11) dh/dt =
12) dh/dt =
13) dx/dt = 40 mi/min
14) dp/dt = 50 lbs/cm2/sec