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Thread: Transformation of functions and their graphs

  1. #1
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    Transformation of functions and their graphs

    A ballet dancer jumps in the air. The height, $\displaystyle h(t)$, in feet, of the dancer at time $\displaystyle t$, in seconds since the start of the jump, is given by$\displaystyle ^4$.

    $\displaystyle h(t) = -16t^2 + 16Tt$

    where $\displaystyle T$ is the total time in seconds that the ballet dancer is in the air.

    Show that the time, $\displaystyle T$, that the dancer is in the air is related to $\displaystyle H$, the maximum height of the jump by the equation:

    $\displaystyle H = 4T^2$

    Thanks a whole bunch!
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  2. #2
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    Hello hydride
    Quote Originally Posted by hydride View Post
    A ballet dancer jumps in the air. The height, $\displaystyle h(t)$, in feet, of the dancer at time $\displaystyle t$, in seconds since the start of the jump, is given by$\displaystyle ^4$.

    $\displaystyle h(t) = -16t^2 + 16Tt$

    where $\displaystyle T$ is the total time in seconds that the ballet dancer is in the air.

    Show that the time, $\displaystyle T$, that the dancer is in the air is related to $\displaystyle H$, the maximum height of the jump by the equation:

    $\displaystyle H = 4T^2$

    Thanks a whole bunch!
    The velocity of the dancer at time $\displaystyle t$ is given by

    $\displaystyle v = h'(t) = -32t+16T$

    The dancer is at the maximum height when $\displaystyle v=0$; i.e. when $\displaystyle t = \tfrac12T$

    So the maximum height H is given by $\displaystyle H=h(\tfrac12T)=-16(\tfrac12T)^2+8T^2$

    $\displaystyle \Rightarrow H = 4T^2$

    Grandad
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