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Math Help - Optimisation - surface area

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    Optimisation - surface area

    A square-based rectangular prism has a total surface area of 2400cm^2. What are the dimensions of the prism if its volume is a maximum?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by bubbles73 View Post
    A square-based rectangular prism has a total surface area of 2400cm^2. What are the dimensions of the prism if its volume is a maximum?
    Let x represent the sides of the base of the square-based rectangular prism, and let h be the height of the prism.

    We are told that the surface area is 2400 cm^2.

    The equation we will use to define surface area will be 2400=2x^2+4xh

    Solving for h, we get h=\frac{600}{x}-\tfrac{1}{2}x

    Since V=x^2h, we can now substitute the expression we came up with into h:

    V=x^2\left(\frac{600}{x}-\tfrac{1}{2}x\right)=600x-\tfrac{1}{2}x^3

    Differentiate and then set equal to zero.

    V'=600-\tfrac{3}{2}x^2\implies 600=\tfrac{3}{2}x^2\implies 400=x^2\implies x=\pm20

    We take the positive value, since length can't be negative.

    Now this implies that h=\frac{600}{20}-\tfrac{1}{2}(20)=\color{red}\boxed{20}

    Thus, the dimensions of the prism are \color{red}\boxed{20~cm~\times~20~cm~\times~20~cm}

    This prism is actually a cube.

    Does this make sense?

    --Chris
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