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Math Help - Finding a rate of change, and mean value theorem

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    Finding a rate of change, and mean value theorem

    A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building.


    And


    Find numbers that satisfy the conclusion of the mean value theorem for f(x) = x/(x-1) on [2, 5]

    Thanks to anyone that answers the post!
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    Quote Originally Posted by Gfreak View Post
    A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building.


    And


    Find numbers that satisfy the conclusion of the mean value theorem for f(x) = x/(x-1) on [2, 5]

    Thanks to anyone that answers the post!
    For the first one use similar triangles.

    Let y be the height of the shadow on the wall and let x be the distance from the light then

    \frac{y}{2}=\frac{12}{x} \implies \frac{dy}{dt}=-\frac{24}{x^2}\frac{dx}{dt}

    For the 2nd one just solve the equaiton.

    \frac{f(5)-f(2)}{5-2}=f'(c) for c.
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