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Math Help - Constructing a Closed Box

  1. #1
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    Constructing a Closed Box

    A closed box with a square base is required to have a volume of 10 cubic feet.

    (a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base.

    (b) How much material is required for a base 1 foot by 1 foot?

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  2. #2
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    Hello, Magentarita!

    A closed box with a square base is required to have a volume of 10 ft³

    (a) Express the amount A of material used to make such a box
    as a function of the length x of a side of the square base.
    Code:
             *-----*
            /     /|
           /     / | y
          *-----*  |
          |     |  *
        y |     | /
          |     |/ x
          *-----*
             x

    The volume is 10 ft³: . V \:=\:x^2y \:=\:10 \quad\Rightarrow\quad y \:=\:\frac{10}{x^2} .[1]

    The surface area is: . A \;=\;2x^2 + 4xy .[2]

    Substitute [1] into [2]: . A \;=\;2x^2 + 4x\left(\frac{10}{x^2}\right)

    Therefore: . \boxed{A \;=\;2x^2 + \frac{40}{x}}



    (b) How much material is required for a base 1 ft by 1 ft?
    If x = 1\!:\;\;A \;=\;2(1^2) + \frac{40}{1} \;=\;\boxed{42\text{ ft}^2}

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  3. #3
    MHF Contributor
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    Wonderfully done

    Quote Originally Posted by Soroban View Post
    Hello, Magentarita!

    Code:
             *-----*
            /     /|
           /     / | y
          *-----*  |
          |     |  *
        y |     | /
          |     |/ x
          *-----*
             x
    The volume is 10 ft³: . V \:=\:x^2y \:=\:10 \quad\Rightarrow\quad y \:=\:\frac{10}{x^2} .[1]

    The surface area is: . A \;=\;2x^2 + 4xy .[2]

    Substitute [1] into [2]: . A \;=\;2x^2 + 4x\left(\frac{10}{x^2}\right)

    Therefore: . \boxed{A \;=\;2x^2 + \frac{40}{x}}


    If x = 1\!:\;\;A \;=\;2(1^2) + \frac{40}{1} \;=\;\boxed{42\text{ ft}^2}
    I love your answers to my questions. I will be taking calculus 1 next semester. I hope you are there for me.
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