A toy rocket is shot vertically into the air from a launching pad 5 feel above the ground with an initial velocity of 80 feet per second. The height h, in feet, of the rocket above the ground as t seconds after launch is given by the function h(t)=-16t^2+80t+5. How long will it take the rocket to reach its maximum height? What is the maximum height?

CAN ANYONE HELP ME ASAP!!!!
Thank you in advance

\(h(t)=-16t^2+80t+5\\\\t_{max}-time\ for\ a\ maximum\ height\\\\t_{max}=- \frac{80}{2\cdot(-16)} = \frac{80}{32} =2.5\ [s]\\\\h_{max}-the\ maximum\ height\ above\ the\ ground\\\\h_{max}=h(2.5)=-16\cdot2.5^2+80\cdot2.5+5=-16\cdot6.25+200+5=\\.\ \ \ \ \ \ =-100+205=105\\\\h_{max\ rocket}-the\ maximum\ height\ of\ a\ toy\ rocket\\\\h_{max\ rocket}=105-5=100\ [ft]\\\\Ans.\ t_{max}=2.5\ second,\ \ h_{max\ rocket}=100\ feet.\)

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carlosego carlosego · Answer 2 · 2018-08-10T12:18:35-04:00

For this case we have the following function:

\(h (t) = - 16t ^ 2 + 80t + 5
\)

To find the time when it reaches its maximum height, what we must do is to derive the function.

We have then:

\(h '(t) = - 32t + 80
\)

We set zero and clear the time:

\(-32t + 80 = 0

32t = 80
\)

\(t =\frac{80}{32}

t = 2.5 s
\)

Then, we evaluate the time obtained for the function of the height.

We have then:

\(h (2.5) = - 16 * (2.5) ^ 2 + 80 * (2.5) +5

h (2.5) = 105 feet
\)

Answer:

It will take the rocket to reach its maximum height:

\(t = 2.5 s
\)

the maximum height is:

\(h (2.5) = 105 feet\)