AP CALCULUS AB
Name: _____________________________
WORKSHEET 4.1-3
4 3
r
3
Surface area of a sphere: SA  4r 2
Volume of a sphere:
V
Volume of a cone:
V
1 2
r h
3
1. Air is being pumped into a spherical balloon at the rate of 7 cubic centimeters per second. What is the rate of change
of the radius at the instant the volume equals 36 cubic centimeters?
2. Oil spilled from a tanker spreads in a circle whose circumference increases at a rate of 60 feet per second. How fast is
the area increasing when the circumference is 180π feet?
3. Gas is escaping from a spherical balloon at the rate of 2 cubic feet per minute. How fast is the surface area shrinking
when the radius is 12 feet?
4. A point P is moving along the graph of
per second. How fast is x changing?
y  x 3  17
. When P is at (2,5) , y is increasing at a rate of 2 units
5. The area of a rectangle is increasing at a rate of 2 in2/min when it is 32 in2. If the length is 4 in and the width is
increasing by 2 in/min, how is the length changing?
6. An airplane passes overhead traveling at 1000 feet per second at an altitude of 10000 feet. The airplane maintains its
altitude as you watch it pass over. At what rate is the plane moving away from you when it has traveled 24000 feet?
7. Flight 217 is 200 km west of Flight 328. Flight 217 flies west at 500 kph and flight 328 flies south at 600 kph. How
fast are the planes separating 90 minutes later?
8. Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being
pumped into the tank at a constant rate. The tank has a height of 6 meters and the diameter is 4 meters. If the water
level is rising at a rate of 20 cm/min when the height of the water is 2 meters, find the rate at which water is being
pumped into the tank.