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Math Help - optimization

  1. #1
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    optimization

    an open rectangular storage shelter is 4 ft. deep, consisting of two vertical sides and a flat roof, is attached to an existing structure. the flat roof is made of aluminum and costs 5 dollars per square foot. the two sides are made of plywood costing 2 dollars per square foot. if the budget for the project is 400 dollars, determine the dimensions of the shelter which will maximize the volume.

    thanks
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  2. #2
    Super Member

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    Hello, doctorgk!

    Did you make a sketch?


    An open rectangular storage shelter is 4 ft. deep, consisting of
    two vertical sides and a flat roof, is attached to an existing structure.
    The flat roof is made of aluminum and costs 5 dollars per square foot.
    The two sides are made of plywood costing 2 dollars per square foot.
    If the budget for the project is $400, determine the dimensions
    of the shelter which will maximize the volume.
    Code:
                     x
              * - - - - - - *
           4 /             /|
            /             / |
           /             /  |y
          * - - - - - - *   |
          |   |         |   |
          |   *         |   *
         y|  /          |  /
          | /           | / 4
          |/            |/
          *             *

    The width of the shelter is x feet.
    The height of the shelter is y feet.
    The depth of the shelter is 4 feet.

    The roof's area is 4x ft².
    At $5/ft², its cost is: 20x dollars.

    The sides have an area of 2(4y) = 8y ft².
    At $2/ft², they will cost: 2(8y) = 16y dollars.

    The total cost is: . 20x + 16y \:=\:400
    . . Solve for y\!:\;\;y \:=\:25 - \frac{5}{4}x .[1]


    The volume of the shelter is: . V \;=\;4xy .[2]

    Substitute [1] into [2]: . V \;=\;4x\left(25 - \frac{5}{4}x\right)

    Hence: . \boxed{V \;=\;100x - 5x^2} is the function you must maximize.

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