AP CALCULUS AB
Name: _____________________________
WORKSHEET 4.1-1
Example Problems
1. The radius of a circle is increasing at a rate of 2 in/min. Find the rate of change of the area when r = 6 in and
r = 24 in.
2. A spherical balloon is being filled (it retains its shape). If air is being pumped in at a rate of 5 cm3/sec, how rapidly
is the radius changing when the radius is 15 cm?
3. A 25 ft ladder is leaning against the side of a house. The base is pulled away from the house at a rate of 2 ft/sec.
How fast is the top of the ladder moving down the side of the house when the base of the ladder is 7 ft from the
house?
4. A child is flying a kite. If the kite is 90 ft above the child’s hand level and the wind is blowing the kite on a
horizontal course at 5 ft/sec, how fast is the child letting out the cord when 150 ft of the cord is out?
5. A small balloon is released at a point 150 ft away from an observer who is on level ground. If the balloon rises at
a rate of 8 ft/sec, how fast is the distance from the observer to the balloon increasing when the balloon is 50 ft high?
6. An automobile traveling at a rate of 30 ft/sec is approaching an intersection. When the automobile is 120 ft from the
intersection, a truck traveling at the rate of 40 ft/sec crosses the intersection. The automobile and truck are on roads
that are perpendicular to each other. How fast are the automobile and truck separating 2 seconds after the truck
leaves the intersection?
7. A water tank has the shape of an inverted circular cone with a base radius of 2 m and a height of 4 m. If water is
being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is
3 m deep.
Practice Problems
1. A kite is flying at a height of 40 ft. A child is flying it so that it is moving horizontally at a rate of 30 ft/sec. If the
string is taut, at what rate is the string being let out when the length of the string released is 50 ft?
2. A ladder 7 meters long is leaning against a wall. If the bottom of the ladder is pushed horizontally toward the wall at
1.5 m/sec, how fast is the top of the ladder sliding against the wall when the bottom of the ladder is 2 meters from the
wall?
3. A rocket is launched and travels straight up at a constant velocity of 750 mph. An observer is 2.5 miles away
watching the launch. How fast is the distance from the observer to the rocket increasing when the rocket is 1 mile
high?
4. At 1pm, a plane travels over CB East traveling south at a constant velocity of 450 mph. At 1:30pm, another plane
passes over CB East traveling east at 600 mph. At 3pm, how rapidly will the distance between the planes be
increasing?
5. Water is pouring into a conical cistern at the rate of 8 ft3/min. If the height of the cistern is 12 ft and the radius of its
circular opening is 6 ft, how fast is the water level rising when the water is 4 ft deep?
6. A spherical snowball is being made so that its volume is increasing at the rate of 8 cubic feet per minute. Find the
rate at which the radius is increasing when the snowball is 4 feet in diameter?
7. The area of a circle is increasing at the rate of 20 square inches per second. Find the rate at which the radius of the
circle is increasing when the diameter is 8 inches?