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.

Step-by-step explanation:

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

`(d)/(dt)h(t)=-32t+16`

Now,

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

Therefore,

`\text{ At }t=(1)/(2),\text{ }(d^2)/(dt^2)=-32<0`

Hence the given function is maximum at

`t=(1)/(2)`

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

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

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

`(1)/(2)\text{ sec.}`

Final answer:

Hence the final answer is

`(1)/(2)\text{ sec.}`