Example 1: A pebble is dropped into a calm pond, causing ...
AP Calculus AB Name: ________________________
Chapter 2: Related Rates Date: ____
Example 1: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius “r” of the outer ripple is increasing at a constant rate of 1 ft/sec. When the radius is 4 ft, at what rate is total area “A” of the disturbed water changing?
Example2: Air is pumped into a spherical balloon at rate of 4.5 in3/min. Find the rate of change of the radius of the balloon when the radius is r = 2 in.
Example 3: A cone-shaped paper cup is being filled with water at the rate of 3 cm3/sec. The height of the cup is double the radius of the base. How fast is the water level rising when the level is 4 cm?
Example 4: A 13-foot ladder leans against a vertical wall. If the bottom of the ladder is slipping away from the base of the wall at a rate of 2 ft/sec, how fast is the top of the ladder moving down the wall when the bottom of the ladder is 5 ft from the base? At what rate is the angle that the ladder makes with the wall changing?
Example 5: An airplane is flying at an elevation of 6 miles on a flight path that is directly over a radar tracking station. Let “s” represent the air distance (in miles) between the station and the plane. If “s” is decreasing at a rate of 400 mph, when s = 10 miles, what is the velocity of the plane. (Assume the plane is traveling at a consistent elevation).
Example 6: A television camera at ground level is filming the lift-off of a rocket that is rising vertically according to the function s(t) = 50t2, where “s” is measured in feet, and “t” is measured in seconds. The camera is 2000 feet away from the launch pad, on the ground. Find the rate of change in the angle of elevation of the camera at
10 seconds after lift-off.
Example 7: A conveyor belt unloads sand off the back of a large truck onto a pile. The sand forms the shape of a cone that has a height which is always one fourth of its diameter. If the sand is being dumped at a constant rate of 10 cubic centimeters per second, at what rate is the height of the pile changing when the pile is 2 centimeters high? …when the pile is 5 centimeters high? … when there is 5 million cubic centimeters of sand in the pile (use a decimal approximation for this)?
Example 8: Two Zax leave a cross-road at 1 pm after enjoying a delightful lunch with each other. One is a south-bound Zax, walking at a rate of 3 meters per second. The other is a west-bound Zax, walking at 4 meters per second. At what rate is the distance between them changing (in meters per second) 1 hour later? At what rate is the area of the triangular plot of land (implied in the problem) changing at this time?
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