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Math Help - More Optimization Problem . . .

  1. #1
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    More Optimization Problem . . .

    A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer. How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand to maximize the angle theta subtended at his eye by the painting?)

    I'm confident that I've worked the first part of the problem correctly, so I will just post my work for the part that I am having trouble with.

    theta = a and beta = b

    a = (a + b) - b
    a = tan^-1(h + d/x) - tan^-1(d/x)
    Da/dx = -(h + d)/(x^2 + h^2 + 2dh + d^2) +( )d/(x^2 + d^2)
    => -(h + d)(x^2 +d^2) + (d)(x^2 + h^2 + 2dh + d^2)/(x^2 + h^2 + 2dh + d^2)(x^2 + d^2)

    For some reason, when I try to simplify this it just gets worse. Can someone please help me simplify this part, or just show me an easier way to go about this problem?

    Thanks
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  2. #2
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    Hello, zachb!

    I agree with your set-up . . . then I went off on a different path.


    A painting in an art gallery has height h and is hung
    so that its lower edge is a distance d above the eye of an observer.
    How far from the wall should the observer stand to get the best view?
    (In other words, where should the observer stand to maximize
    the angle theta subtended at his eye by the painting?)
    Code:
                                        * B
                                    *   |
                                *       | h
                            *           |
                        *               * C
                    *           *       |
                *       *               | d
            * θ   *                     |
        A * - - - - - - - - - - - - - - * D
                        x

    We have: .θ .= .arctan[(h+d)/x] - arctan(d/x)

    . . which equals: .θ .= .arccot[x/(h+d)] - arccot(x/d)

    . . . . . . . . . . . - 1/(h+d) . . . . . 1/d
    Then: .θ' .= .----------------- + ------------ .= .0
    . . . . . . . . . 1 + [x/(h+d)] . .1 + (x/d)


    Multiply the first fraction by (h+d)/(h+d), the second by d/d

    . . . . . . . . . . . . . . -(h+d) . . . . . . d
    . . and we have: .--------------- + ---------- .= .0
    . . . . . . . . . . . . .(h+d) + x . . d + x


    Clear denominators: .-(h+d)(x + d) + d[x + (h+d)] .= .0

    Expand (partially): .-(h+d)x - d(h+d) + dx + d(h+d) .= .0

    . . and we have: .dx - (h+d)x .= .d(h+d) - d(h+d)

    Factor: .[d - (h+d)]x .= .d(h+d)[d - (h+d)]

    . . and we have: .-hx .= .-hd(h+d) . . x .= .d(h+d)

    . . . . . . . . . . . . . _______
    Therefore: .x .= .√d(h + d)



    Someone check my work . . . please!
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  3. #3
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    We have: .θ .= .arctan[(h+d)/x] - arctan(d/x)

    . . which equals: .θ .= .arccot[x/(h+d)] - arccot(x/d)
    This step seems a litte strange to me. Why did you go from arctan to arccot?
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