Question

A ball is dropped from an initial height of 22 feet and bounces off the ground repeatedly. The function h estimates that the maximum height reached after each time the ball hits the ground is 85% of the maximum height reached where after the previous time the ball hit the ground. Which equation defines h, h(n) is the estimated maximum height of the ball after it has hit the ground n times and n is a whole number greater than 1 and less than 10?

A) h(n)=22(0.22)ⁿ
B) h(n)=22(0.85)ⁿ
C) h(n)=85(0.22)ⁿ
D) h(n)=85(0.85)ⁿ

Answer 1

The equation that defines the estimated maximum height of the ball after it has hit the ground n times is h(n) = 22(0.85)^n. This is an exponential decay function based on the initial height of 22 feet and the bounce height reducing to 85% of the previous height with each bounce.

Step-by-step explanation:

The student's question revolves around constructing a mathematical model that can define the maximum height reached by a ball after it bounces n times. The ball starts at a height of 22 feet, and after each bounce, it reaches 85% of the height from the previous bounce. Therefore, the function should represent an exponential decay based on these parameters.

To find the function, h, that describes the height after n bounces, we use the initial height as the starting value and multiply it by 0.85 raised to the power of n because the ball reaches 85% of the height of the previous bounce with each bounce. This gives us h(n) = 22(0.85)^n, which corresponds to option B from the provided choices. This function successfully models the behavior of the bouncing ball over time as it follows an exponential decay pattern.