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Math Help - Another optimazation problem

  1. #1
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    Another optimazation problem

    a 10 ft display sign is 10 feet off the ground. Find the distance from the sign of a point on the ground where the subtended angle of the sign is a maximum.
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  2. #2
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    Re: Another optimazation problem

    Let x be the optimal distance from the sign, \alpha be the subtended angle of the sign and \beta be the angle formed by the ground and the ray from the optimal viewpoint to the bottom of the sign. Then \tan\beta=10/x and \tan(\alpha+\beta)=20/x. Using the formula for the tangent of difference, you can express \tan((\alpha+\beta)-\beta)=\tan\alpha through x. Since \tan is a monotonic function on [0,\pi/2), the maximum point of \tan(\alpha(x)) is also the maximum point of \alpha(x).
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