Calculus

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A company is designing shipping crates and wants the volume of each crate to be 4 cubic feet, and the crate's base to be a square between 1 and 1.5 feet per side. If the material for the bottom costs $5, the sides $3 and the top $1 per square foot. What dimensions will give the minimum cost?

  • Calculus -

    ______
    /_____/| $5 bottom
    || || $1 Top
    || || H $3 Side
    ||____||
    |/____|/S
    S

    A = S^2 * H Cost will be related
    4 = S^2 * H with Surface Area
    H = 4 / S^2 So our equation is...

    SA = 2S^2 + 4SH
    Cost = 5S^2 + S^2 + 4*3*S(4/S^2)
    Cost = 6S^2 + 48/S

    Now we differentiate Cost

    Cost' = 12S - 48/S^2

    Now we must determine where the derivative of cost is equal to 0

    0 = 12S - 48/S^2
    48/S^2 = 12S
    48 = 12S^3
    S = 4^(1/3)

    So a critical number will occur at the cube root of 4. Do a number line analysis.

    Domain: [1,1.5]
    -- -- ++
    |----------|-----------|-------
    1 1.5 4^(1/3)

    The local minimum occurs at 4^(1/3), which is not in the domain situation. Instead, use 1.5 for the value of S.

    H = 4/S^2
    H = 4/[3/2]^2
    so H = 16/9

    Answer: A 1.5 x 1.5 x 16/9 box will minimize the cost.

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