27. Your friend drops a rubber ball from 4 ft. You notice that its rebound is 32.5
in. on the first bounce and 22 in. on the second bounce.
a. What exponential function would be a good model for the height of the
ball?
b. How high will the ball bounce on the fourth bounce?
a. We can model the height of the ball using the exponential function h(n) = h0 * r^n, where h0 is the initial height, r is the rebound ratio, and n is the bounce number.
From the information given:
h(0) = 4 ft = 48 in
h(1) = 32.5 in
h(2) = 22 in
Using the values for the first two bounces to find the rebound ratio:
32.5 = 48 * r
r = 32.5 / 48 = 0.677
Now we can write the function for the height of the ball:
h(n) = 48 * 0.677^n
b. To find the height on the fourth bounce, we calculate h(3):
h(3) = 48 * 0.677^3 ≈ 20.4 in
Therefore, the ball will bounce approximately 20.4 inches on the fourth bounce.