Download 5.16 Related Rates

5.16 Related Rates – Part 3
AP Calculus Notes
Name: _______________________________________________
Hour: _______
1. A water tank has the shape of an inverted circular cone with base radius of 2 meters
and height of 4 meters. If water is being pumped into the tank at a rate of 2 m 3 / min , find
1
the rate at which the water level is rising when the water is 3 meters deep. V  r 2 h
3
2. You are videotaping a race from a stand 132 ft from the track, following a car that is
moving at 264 ft/sec. About how fast will your camera angle be changing one second
after the car passes in front of you?
3. Two trucks convoys leave a depot. Convoy A travels east at 40 mph and convoy B
travels north at 30 mph. How fast is the distance between the convoys changing six
minutes later, when convoy A is 4 miles from the base and convoy B is 3 miles from the
base?
4. A large spherical balloon is being inflated at the rate of 100 cubic feet per minute. At
4
what rate is the radius increasing when the radius is 10 feet ( V  r 3 )?
3
5. Two commercial jets at 40,000 ft are flying at 520 mi/hr along straight-line courses
that cross at right angles. How fast is the distance between the airplanes closing when
airplane A is 5 miles from the intersection point and airplane B is 12 miles from the
intersection point?
6. If a snowball melts so that its surface area decreases at a rate of 1cm²/min, find the
rate at which the diameter decreases when the diameter is 10 cm. ( SA  4r 2 )
7. The length of a rectangle is increasing at 3 ft/min and the width is decreasing at 2
ft/min. When the length is 50 ft and the width is 20 ft, is the area of the rectangle
increasing or decreasing? At what rate?
4
8. The radius of a sphere is increasing at a constant rate of 0.5 in/sec. V  r 3
3
(a)
When the radius of the sphere is 15 inches, at what rate is the volume of the
sphere changing?
(b)
When the volume and radius of the sphere are changing at the same rate, what is
the radius of the sphere?
9. Suppose you are drinking pop from a conical paper cup. The cup has a diameter of 8
cm and a depth of 10 cm. As you suck on the straw, the pop leaves the cup at the rate of
7 cm³/sec. At what rate is the level of liquid in the cup changing when the liquid is 6 cm
1
deep? V  r 2 h
3
10. Some of Miss Thompson’s students are trying to get to class on time. A police
officer is hovering in a helicopter 1000 feet directly above the road using a radar gun to
determine the speed at which they are traveling. When the students are 2000 feet (actual
distance) from the helicopter and moving away, the radar gun registers 85 feet per
second. The speed limit on the highway is 95 feet per second (65 mph). Were they
speeding? Assume that the road is straight and level.