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Math Help - Prism

  1. #1
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    Prism

    A rectangular prism, of which the base is a square, has a diagonal of length d, and the total area of all its faces is b. In terms of b and d, what is the total length of all its edges?
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  2. #2
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    Quote Originally Posted by atreyyu View Post
    A rectangular prism, of which the base is a square, has a diagonal of length d, and the total area of all its faces is b. In terms of b and d, what is the total length of all its edges?
    Let x = side of square base,
    and y = height of prism
    And assume that the prism is a right rectangular prism, or the the base and the height are perpendicular.

    In the square base:
    diagonal = d .....given
    So, by Pythagorean theorem, d^2 = x^ +x^2
    x^2 = (d^2) /2
    x = d / (sqrt(2))

    Area of all the faces of the prism:
    A = 2(x^2) +4(x*y) = b .....given
    2x^2 +4xy = b
    4xy = b -2x^2
    y = (b -2x^2) / (4x)
    y = [b -2(d^2 / 2)] / [4(d / sqrt(2)]
    y = [b -d^2] / [2sqrt(2) *d]
    Simplify further if you like...

    Therefore, the sum of all the edges of this prism is
    = 8x +4y
    = [8(d / sqrt(2)] +[4(b -d^2) / (2sqrt(2) *d)]
    = [4sqrt(2) *d] + [2sqrt(2) *(b -d^2)/d]
    = 2sqrt(2)*[2d +(b -d^2)/d]
    = 2sqrt(2)[(2d^2 +b -d^2) / d]
    = 2sqrt(2)[(b +d^2)/d] -----------------answer.
    Last edited by ticbol; August 4th 2008 at 04:51 PM. Reason: 4y, not 47, for the sum of edges.
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