Name:
Block:
Date: __________ AP Calculus
Cont Sec 4.6 Related Rates
A QUESTION FROM 1985 AP EXAM #31
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π r 2h
3
1
3
If the radius and the height BOTH increase at a constant rate of cm/s, at what rate, in cm / s , is the
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volume increasing when the height is 9 cm and the radius is 6 cm?
The volume of a cone of radius r and height h is given by V =
RELATED RATE PROBLEMS INVOLVING DISTANCE:
Ex 1. A taxi drives 3 km east, and then turns north, travelling at 50 km/h. How fast is the distance
between the cab and the starting point increasing when the taxi has driven 4 km north?
N
E
Ex 2. A bug is walking away from a wall 40 m high at a rate of 3 m/min. At what rate is the distance
from the top of the wall changing when it is 30 m from the wall.
Ex 3. A right triangle has a hypotenuse of constant length 10 cm. One leg of the triangle increases at
a rate of 2 cm/sec. When it is 6 cm long what is the rate at which the other leg is decreasing.
Ex 4. A ladder 13 m long is resting flat against a building. The bottom of the ladder is being pulled
along the ground a constant rate of 3 m/min. How fast is the height of the top of the ladder decreasing
after 4 minutes.
Ex 5
A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that
has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the
intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between
them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of
measurement, what is the speed of the car?
N
E
Ex 6
A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off
π
point. At the moment the range finder’s elevation angle is , the angle is increasing at the rate of 0.14
4
rad/min. How fast is the balloon rising at that moment?
Ex 7. A man is walking away from a street light at a rate of 5 ft/sec. If the man is 6 ft tall and the
light is 15 ft high, how fast is the man's shadow lengthening when he is 10 ft from the base of the street
light.
QUESTION FROM 1991 AP EXAMINATION
8. A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are
50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to
point B, is illuminated by a spotlight 70 feet above point A, as shown in the diagram.
a) How fast is the shadow of the tightrope walker's feet moving long the ground when she is midway
between the Tee buildings? (Indicate units of measure.)
b) How far from point A is the tightrope walker when the shadow of her feet reaches the base of the
Tee building? (Indicate units of measure.)
c) How fast is the shadow of the tightrope walker's feet moving up the wall of the Tee Building when
she is 10 feet from point B? (Indicate units of measure.)