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calculus

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Find the minimum cost of producing a can that contains 1000mm^3 if the cost of the metals is 2 cents per mm^2 and the cost to weld the seams is 1 cent per mm. Assume three welds;one around the top, one around the bottom, and one down the side.

  • calculus -

    let the radius of the can be r mm
    and its height be h mm

    area = 1 rectangle + 2 circles
    = 2pi(r)(h) + 2pi(r^2)


    seems length = 2 circuferences + height of can
    = 2(2pi(r)) + h
    = 4pi(r) + h

    so Cost = 2[2pi(r)(h) + 2pi(r^2)] + 1[4pi(r) + h]

    but pi(r^2)h = 1000
    so h = 1000/(pi(r^2))

    this should give you a good start, sub h back into the Cost equation,
    find the derivative of Cost,
    then solve for r

    sub back into Cost equation to find minimum cost

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