Calculus

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A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder with a hempishere at each end. The cost per square foot of constructing the end pieces is twice that of contructing the cylindrical piece. If the desired capacity of the tank is 144 pi cubic feet, what dimensions will minimize the cost of construction?

  • Calculus -

    The two hemispheres at the ends will make up one complete sphere
    Let the radius of the sphere be r ft, and let the height of the cylindrical part be h
    Volume = cylinder + sphere
    πr^2 h + (4/3)πr^3 = 144π
    3πr^2 h + 4πr^3 = 432π
    3r^2 h + 4r^3 = 432
    h = (432 - 4r^3)/(3r^2)

    The cost of production is dependent on the surface area of material used

    SA = 4π^2 + 2πrh
    = 4πr^2 + 2π(432 - 4r^3)/(3r^2)
    = 4πr^2 + 288π/r^2 - (8/3)πr

    d(SA)/dr = 8πr - 576π/r^3 - 8π/3 = 0 for a min of SA
    8r - 576/r^3 - 8/3 = 0
    times 3r^3
    24r^4 -1728 - 8r^3 = 0
    3r^4 - r^3 - 216 = 0


    At this point I "cheated" a bit and ran it through Wolfram
    http://www.wolframalpha.com/input/?i=3r%5E4+-+r%5E3+-+216+%3D+0
    for a real solution of r=3, r = appr-3 and 2 complex roots,

    so the only usable answer is r = 3
    the h = (432 - 108)/27 = 12

    So the cylinder part should be 12 ft long and the radius of each of the semispheres should be 3 ft

  • Calculus -

    remember the sphere surface is twice as expensive as the cylinder which has no compound curvature so is easy to roll.

    c = 2(4 pi r^2) + 1(2 pi r h)
    c= 8 pi r^2 + 2 pi r h

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