Mathematics

Question

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

2 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
    [/tex]

    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
    [/tex]

    We set zero and clear the time:

    [tex] -32t + 80 = 0

    32t = 80
    [/tex]

    [tex] t =\frac{80}{32}

    t = 2.5 s
    [/tex]

    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
    [/tex]

    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]