Final answer:
To find the time when the rocket is 112 feet above the ground, we solve the quadratic equation h(t) = -16t^2 + 128t by setting it equal to 112 and factoring. The rocket is at 112 feet at two times: 1 second (on the way up) and 7 seconds (on the way down).
Step-by-step explanation:
To determine after how many seconds the rocket will be 112 feet above the ground, we need to solve the equation given by the rocket's height as a function of time, h(t) = -16t^2 + 128t. We want to find the time t when the height is 112 feet. Therefore, we set the equation equal to 112 and solve for t.
112 = -16t^2 + 128t
This is a quadratic equation that can be rewritten as:
0 = -16t^2 + 128t - 112
Dividing the entire equation by -16 to simplify gives:
0 = t^2 - 8t + 7
Factoring this equation, we get:
0 = (t - 7)(t - 1)
So, the possible values for t are t = 7 seconds and t = 1 second. Both values are the times when the rocket is at a height of 112 feet, once on the way up and once on the way down.