Question

Hello,Can you help me find the maximum height of the arrow?

Answer 1

400 ft

Step-by-step explanation

We have that the height of the arrow t seconds after it was shot is:

`h(t)=-16t^2+160t`

We want to find the maximum height of the arrow. Since the arrow follows a parabolic path, the maximum height is reached when the vertical speed of the arrow is 0 - this is because the arrow slows down as it goes up, then stops for a moment and speeds back down.

The velocity of the arrow is the derivative of the height:

`v=h^(\prime)=-32t+160`

To find the time we solve:

`0=-32t+160`

Add 32t to both sides:

`\begin{gathered} 0+32t=-32t+32t+160 \\ 32t=160 \end{gathered}`

And divide both sides by 32:

`\begin{gathered} (32t)/(32)=(160)/(32) \\ t=5 \end{gathered}`

Now we replace t = 5 into the height equation:

`h(5)=-16\cdot5^2+160\cdot5`

And solve:

`\begin{gathered} h(5)=-16\cdot25+800 \\ h(5)=-400+800 \\ h(5)=400 \end{gathered}`

The maximum height of the arrow is 400 ft