MATH 151 - Calculus: Related Rates (Section 2.7)
1. A spherical balloon is inflated so that its radius increases at a rate of 0.25 inches per second. At the
moment when the balloon’s radius is 15 inches, how fast is the volume of the balloon increasing?
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2. Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm 3/sec. At the
moment when the balloon’s diameter is 50 cm, how fast is the radius of the balloon increasing?
3. A 25 foot ladder rests against the outside wall of a building. Assume the bottom of the ladder slides
away from the base of the wall at a rate of 2 ft/sec.
a) How fast is the top of the ladder sliding down the wall at the moment when the bottom of the ladder
is exactly 7 ft from the wall?
b) Consider the triangle formed by the ladder and the building. Using the conditions above, find the rate
at which the area of this triangle is changing when the bottom of the ladder is 7 feet from the wall.
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4. Car A is traveling west at 50 mi/hour and Car B is traveling north at 60 mi/hour. Each car is headed
toward the intersection of the roads they are traveling on. When Car A is 0.3 miles from the intersection,
and Car B is 0.4 miles from the intersection, what is the rate at which the cars are approaching each
other?
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5. An air traffic controller spots two planes on its radar flying at the same altitude. The flight path of the
planes has each plane headed toward the same point (they are flying “perpendicular” to each other).
One plane is 150 miles from the potential point of impact and moving at 450 mi/hour, while the other
plane is 200 miles from the point of impact moving at 600 mi/hour.
a) At what rate is the distance between the planes decreasing?
b) How much time does the air traffic controller have to move one of the planes onto a different flight
path?
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6. A conical water tank has a base radius of 2 meters and a height of 4 meters. Suppose that water is being
pumped into the tank at a rate of 2 m3/min. Find the rate at which the water level is rising within the tank
when the water is 3 meters deep.