AP Calculus AB Exam Question

A baseball is being hit in a straight line towards center field. The height of the baseball above the ground at time t is given by the equation h(t) = -16t^2 + 80t, where h(t) is measured in feet and t is measured in seconds.

At what rate is the baseball's height changing 2 seconds after it is hit?

Answer

To find the rate at which the baseball's height is changing 2 seconds after it is hit, we need to calculate the derivative of the height function h(t) with respect to time (t). This derivative will give us the rate of change of the height.

Using the power rule of differentiation, we can find the derivative of h(t) = -16t^2 + 80t as follows:

$\frac{d}{dt}(-16t^2 + 80t) = -32t + 80$

Therefore, the derivative of h(t) with respect to t is -32t + 80.

To find the rate of change of the height 2 seconds after the baseball is hit, we substitute t = 2 into the derivative expression.

$\frac{d}{dt}(-32t + 80) = -32(2) + 80 = -64 + 80 = 16$

Therefore, at 2 seconds after the baseball is hit, the rate at which its height is changing is 16 feet per second.

Answer: The baseball's height is changing at the rate of 16 feet per second, 2 seconds after it is hit.