calculus

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A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. Building the tank costs $10 per square meter for the base and $5 per square meter for the sides.
a) Write a function C, the cost of constructing the described tank as a function of l(the length) and h(the height).
b) Write a function C, the cost of constructing the described tank as a function of a single variable.
c) What is the cost of the least expensive tank? (show all work)

  • calculus -

    let its length be l m
    and its height by h m
    volume = 4lh
    but 4lh = 36
    h = 9/l

    a) cost = 10(4l) + 5(2lh) + 5(8h)
    = 40l + 10lh + 40h


    b) cost = 10(4l) + 5(2l)(9/l) + 40(9/l
    = 40l + 90 + 360/l ,where l ≠ 0

    c) d(cost)/dl = 40 + 0 - 360/l^2 = 0 for a min of cost
    40 = 360/l^2
    l^2 = 9
    l = √9 = 3

    when l = 3, h = 9/3 = 3
    dimensions for min cost = 3by4 for the base and a height of 3
    cost = 330.00

    test: take a value slightly higher and lower than l = 3

    l = 3.1, cost = 330.13 , higher
    l = 2.9 , cost = 330.14 , higher

    answer looks good

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