Answer :
[tex]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.[/tex]
For this case we have the following function:
[tex] h (t) = - 16t ^ 2 + 80t + 5
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To find the time when it reaches its maximum height, what we must do is to derive the function.
We have then:
[tex] h '(t) = - 32t + 80
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We set zero and clear the time:
[tex] -32t + 80 = 0
32t = 80
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[tex] t =\frac{80}{32}
t = 2.5 s
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Then, we evaluate the time obtained for the function of the height.
We have then:
[tex] h (2.5) = - 16 * (2.5) ^ 2 + 80 * (2.5) +5
h (2.5) = 105 feet
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Answer:
It will take the rocket to reach its maximum height:
[tex] t = 2.5 s
[/tex]
the maximum height is:
[tex] h (2.5) = 105 feet [/tex]