Chapter A, Problem 30E

To determine

To find: The time interval when the ball is at least 32ft height.

Answer to Problem 30E

The time interval is $\left[0,3\right]$.

Explanation of Solution

Given:

The formula for obtaining height is, $h=128+16t-16t^2$.

Calculation:

Since the height of the ball should be 32 ft minimum, $h\ge 32$.

Thus, the equation becomes,

$\begin{array}{l}32\le 128+16t-16t^2\\ 16t^2-16t-96\le 0\end{array}$

Find the roots of the above equation, $16\left(t-3\right)\left(t+2\right)=0$.

The roots are $3\text{ and }-2$. Thus, time interval is, $-2\le t\le 3$.

Since time cannot be negative, the interval of time when the ball is at minimum 2 ft is $\left[0,3\right]$.