Question

how long did it take for Katie to reach her maximum height?

Answer 1

Given:

The height function is

$h(t)=-16t+16t+672$

Required:

We have to find the time didi Katie takes to reach her maximum height.

Explanation:

To find the extremal point we find the derivative of the given function and equal it to zero.

$\frac{d}{dt}h(t)=-32t+16$

Now,

$\begin{gathered} -32t+16=0 \\ \Rightarrow t=\frac{-16}{-32} \end{gathered}$$\Rightarrow t=\frac{1}{2}$$\frac{d^2}{dt^2}h(t)=-32$

Therefore,

$\text{ At }t=\frac{1}{2},\text{ }\frac{d^2}{dt^2}=-32<0$

Hence the given function is maximum at

$t=\frac{1}{2}$

At this value the value of the given function that is the maximum height is

$\begin{gathered} h(\frac{1}{2})=-16(\frac{1}{2})^2+16(\frac{1}{2})+672 \\ \\ \Rightarrow h(\frac{1}{2})=-4+8+672 \end{gathered}$$\Rightarrow h(\frac{1}{2})=676$

Therefore, the time taken by Katie to reach her maximum height is

$\frac{1}{2}\text{ sec.}$

Final answer:

Hence the final answer is

$\frac{1}{2}\text{ sec.}$