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Math Help - Trigonometric Related Rate Question

  1. #1
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    Trigonometric Related Rate Question

    An illuminated billboard 10m tall stands on top of a cliff 12m high. How far from the foot of a cliff should a man stand in order for the sun to subtend the largest possible angle at his eyes (which are 2m above the ground)? How large is the maximum angle?


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  2. #2
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    This is an optimization problem, not related rates.

    Anyway, we have to find the value of x in order to maximize the angle.

    Let tan({\alpha})=\frac{22}{x}

    tan({\beta})=\frac{12}{x}

    So, that {\theta}={\alpha}-{\beta}=tan^{-1}(\frac{22}{x})-tan^{-1}(\frac{12}{x})


    \frac{d\theta}{dx}=\frac{12}{x^{2}+144}-\frac{22}{x^{2}+484}

    Set to 0 and solve for x=2\sqrt{66}\approx{16.25}

    The observer should stand 16.25 m from the cliff.

    Check me out. Easy to err.
    Last edited by galactus; November 24th 2008 at 05:38 AM.
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  3. #3
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    How did you come up with the change in theta with respect to time formula?
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  4. #4
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    Could someone provide some insight as to how that formula was created?
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  5. #5
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    It's the only thing I'm having trouble understanding.
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