Name: ______________________
AP Calculus AB Practice Problems: 4.4 Applications of the Derivative
Part A-You may Not use a calculator.
1.
A spherical balloon is being inflated. Find the volume of the balloon at the
instant when the rate of increase of the surface area is eight times the rate of
increase of the radius of the sphere.
2.
A 13-foot ladder is leaning against a wall. If the top of the ladder is sliding down
the wall at 2 ft/sec, how fast is the bottom of the ladder moving away from the
wall, when the top of the ladder is 5 feet from the ground?
3.
Air is being pumped into a spherical balloon at the rate of 100 cm 2 / sec . How
fast is the diameter increasing when the radius is 5 cm?
4.
A man 5 feet tall is walking away from a streetlight hung 20 feet from the ground
at the rate of 6 ft/sec. How fast is his shadow lengthening?
5.
A water tank in the shape of an inverted cone has an altitude of 18 feet and a base
radius of 12 feet. If the tank is full and the water is drained at the rate of 4
ft 3 / min , how fast is the water level dropping when the water level is 6 feet high?
6.
Two cars leave an intersection at the same time. The first car is going due east at
the rate of 40 mph and the second is going due south at the rate of 30 mph. How
fast is the distance between the two cars increasing when the first car is 120 miles
from the intersection?
8. Find the number in the interval (0,2) such that the sum of the number and its
reciprocal is the absolute minimum.
9. An open box is to be made using a cardboard 8 cm by 15 cm by cutting a square
from each corner and folding the sides up. Find the length of a side of the square
being cut so that the box will have a maximum volume.
10. What is the shortest distance between the point (2, -1/2 ) and the parabola
y  x 2 ?
11. If the cost function C ( x)  3x 2  5x  12, find the value of x such that the average
cost is a minimum.
12. A man with 200 meters of fence plans to enclose a rectangular piece of land using
a river on one side and a fence on the other three sides. Find the maximum area
that the man can obtain.
Part B-Calculators allowed.
13. A trough is 10 meters long and 4 meters wide. The two sides of the trough are
equilateral triangles. Water is pumped into the trough at 1m 3 /min. How fast is
the water level rising when the water is 2 meters high?
10
m
4
m
14. A rocket is sent vertically up in the air with the position function s  100t 2 where
s is measured in meters and t in seconds. A camera 3000 m away is recording the
rocket. Find the rate of change of the angle of elevation of the camera 5 sec after
the rocket went up.
15. A plane lifts off from a runway at an angle of 20 degrees. If the speed of the
plane is 30 mph, how fast is the plane gaining altitude?
16. Two water containers are being used. One container is in the form of an inverted
right circular cone with the height of 10 feet and a radius at the base of 4 feet. The other
container is a right circular cylinder with a radius of 6 feet and a height of 8 feet. If water
is being drained form the conical container into the cylindrical container at the rate of 15
cubic ft/min, how fast is the water level falling in the conical tang when the water level in
the conical tank is 5 feet high? How fast is the water level rising in the cylindrical
container.