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calculus

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A cylindrical container is to hold 20π cm^3. The top and bottom are made of a material that costs $0.40 per cm^2. The material for the curved side costs $0.32 per cm^2.

Find the height in centimeters of the most economical container.

  • calculus -

    let the radius be r
    let the height be h

    V = πr^2 h
    πr^2 h = 20π
    h = 20/r^2

    Cost = .40(2πr^2) + .32(2πrh)
    C = .8πr^2 + .64πr(20/r^2)
    = .8πr^2 + 12.8π/r
    dC/dr = 1.6πr - 12.8π/r^2
    = 0 for a min of C

    1.6πr = 12.8π/r^2
    1.6r^3 = 12.8
    r^3 = 8
    r = 2
    then h = 20/4 = 5

    the cylinder should be 5 cm high and have a radius of 2 cm.

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