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Math Help - Rate of Change

  1. #1
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    Rate of Change

    Simple calculus question that i just cannot get my brain around currently.

    A lamp is 6m directly above a straight path. A man 2m tall walks along the path away from the light at a constant speed of 1 m/s. At what speed is the end of his shadow moving along the path? At what speed is the length of his shadow increasing.

    A full working would be appreciated. But even if just show me the set up that would still be good.
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  2. #2
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    Quote Originally Posted by lukybear View Post
    simple calculus question that i just cannot get my brain around currently.

    A lamp is 6m directly above a straight path. A man 2m tall walks along the path away from the light at a constant speed of 1 m/s. At what speed is the end of his shadow moving along the path?
    Rate of Change-rrtriangle.jpg

    Let r=x+s

    We are given that \frac{dx}{dt}=1 and we want to find \frac{dr}{dt}

    \frac{dr}{dt}=\frac{dx}{dt}+\frac{ds}{dt}

    The sides of the similar triangles are proportional so

    \frac{x+s}{s}=\frac{6}{2}=3

    x+s=3s

    x=2s

    \frac{dx}{dt}=2\frac{ds}{dt}

    \frac{ds}{dt}=\frac{1}{2}\frac{dx}{dt}

    \frac{dr}{dt}=\frac{dx}{dt}+\frac{1}{2}\frac{dx}{d  t}=1.5 \frac{m}{s}

    Quote Originally Posted by lukybear View Post
    at what speed is the length of his shadow increasing.

    A full working would be appreciated. But even if just show me the set up that would still be good.
    We want to find \frac{ds}{dt}
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