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A cylinder shaped can needs to be constructed to hold 400 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

  • Math -

    let the radius be r
    let the height be h
    Volume = πr^2h
    πr^2h = 400
    h = 400/(πr^2)

    cost = different prices x surface areas
    = .03(2πrh) + 2(.06) πr^2
    = .03[2πr(400/πr^2) + 4πr^2]
    = .03[ 800/r + 4πr^2]
    d(cost)/dr = .03[ -800/r^2 + 8πr] = 0 for a min of cost

    800/r^2 = 8πr
    100/π = r^3
    r = 3.169
    h = 400/(π(3.169)^2) = 12.679

    check my arithmetic

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