AP Calculus AB: 7.4.3 The Baseball Problem
This content covers related rates problems involving changing distances, such as how the distance between a runner and second base changes as the runner moves. It also explores implicit differentiation with functions of multiple variables changing over time, demonstrating how to find the rate of change of a composite function.
The Baseball Problem
Related rate problems involve using a known rate of change to find an associated rate of change.
Use implicit differentiation when you cannot write the dependent variable in terms of the independent variable.
A negative value for the derivative means the original function is decreasing.
Key Terms
The Baseball Problem
Related rate problems involve using a known rate of change to find an associated rate of change.
Use implicit differentiatio...
note
As a runner is moving toward first base, the distance between the runner and second base changes.
Suppose a runner is moving...
Suppose you are told that z = x ^2y^ 3, where x and y are changing with time. Suppose also that x is increasing at a constant rate of 3 units / second and that y is decreasing at a constant rate of 2 units / second. What is the rate of change of z with respect to time when x = 5 and y = 7?
2,940 units / second
Sand is flowing out of a hopper at a constant rate of 2/3 cubic feet per minute into a conical pile whose height is always twice its radius. What is the rate of change of the radius of the cone when the cone is 4 feet high?
0.027 feet per minute
The distance between home plate and first base on a baseball diamond is 90 ft. A runner is moving towards first base at 24 ft / sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 60 ft away from first base?
dh/dt=−48/√13 ft/sec
Peter is running on a circular track with a radius of 400 meters. If he is running at a constant rate of 6 meters per second, what is the rate of change of his distance from the center of the circular track?
0 meters / second
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| Term | Definition |
|---|---|
The Baseball Problem |
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note |
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Suppose you are told that z = x ^2y^ 3, where x and y are changing with time. Suppose also that x is increasing at a constant rate of 3 units / second and that y is decreasing at a constant rate of 2 units / second. What is the rate of change of z with respect to time when x = 5 and y = 7? | 2,940 units / second |
Sand is flowing out of a hopper at a constant rate of 2/3 cubic feet per minute into a conical pile whose height is always twice its radius. What is the rate of change of the radius of the cone when the cone is 4 feet high? | 0.027 feet per minute |
The distance between home plate and first base on a baseball diamond is 90 ft. A runner is moving towards first base at 24 ft / sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 60 ft away from first base? | dh/dt=−48/√13 ft/sec |
Peter is running on a circular track with a radius of 400 meters. If he is running at a constant rate of 6 meters per second, what is the rate of change of his distance from the center of the circular track? | 0 meters / second |
As the second hand of a clock moves around the dial, it sweeps out a sector with a constantly increasing area. Pablo’s clock has a second hand that is 5 centimeters long. How fast is the area of the sector that is swept out by the second hand increasing? | 1.3 square centimeters per second |