diffirential calculus

a builder intends to construct a storage shed having a volume of 900ft^3, a flat roof and a rectangular base whose width is three-fourths the length. the cost per square foot of the materials is 4000.00 for the floor,6000.00 for the sides and 3000.00 for the roof. what dimension will minimize the cost?

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  1. I like to avoid fractions if possible, so let the
    width be 3x
    and the length be 4x , (notice that 3x/4x) = 3/4)
    let the height be h
    V= (3x)(4x)(h)
    h = 900/(12x^2) = 75/x^2

    Cost = C = 4000(top) + 6000(sides) + 3000(roof)
    = 4000(12x^2) + 6000(8xh+6xh) + 3000(12x^2)
    = 84000x^2 + 84000xh
    = 84000x^2 + 84000x(75/x^2
    = 84000x^2 + 84000(75)/x
    = 84000(x^2 + 75/x)

    d(Cost)/dx = 84000(2x - 75/x^2) = 0 for a min of C
    2x = 75/x^2
    x^3 = 75/2 = 37.5
    x = (37.5)^(1/3) = 3.347

    width = 3(3.347) = 10.04
    length = 4(3.347) = 13.39
    height = 75/(3.347)^2 = 6.69 (all in feet)

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