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Aldrichyang
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Tyro
Asked: 2021-04-08T14:26:39+00:00 In: ,

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Engineers are designing a box-shaped aquarium with a square bottom and an open top. The aquarium must
hold 864 cubic ft of water. What dimensions should they use to create an acceptable aquarium with the least
amount of glass?

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1 Answer

  2. acedstud

    Let the lengths of the sides of the square bottom be x and the height of the aquarium be h, then the volume, V, of the aquarium is given by the area of the base times the height.

    That is, V=x^2h=864 cubic ft. Making h the subject of the formula, we have h=\dfrac{864}{x^2}.

    Now, for the amount of glass used to be minimum, the surface area must be minimum.

    The surface area of the aquarium is given by the sum of the square bottom and the 4 sides, since the aquarium has an open top.

    The area of the bottom top is x^2 and the area of each of the 4 sides is xh. Thus, the surface area of the aquarium is S=x^2+4xh. Substituting for h gives S=x^2+4x\cdot\dfrac{864}{x^2}=x^2+\dfrac{3456}{x}.

    For the surface area to be minimum, \dfrac{dS}{dx}=0\Rightarrow2x-\dfrac{3456}{x^2}=0

    Multiplying through by x^2 gives 2x^3-3456=0\Rightarrow2x^3=3456

    Dividing through by 2 gives x^3=\dfrac{3456}{2}=1728

    Taking the cube root of both sides gives x=\sqrt[3]{1728}=12.

    Now,since h=\dfrac{864}{x^2}, we have h=\dfrac{864}{12^2}=\dfrac{864}{144}=6.

    Therefore, the dimensions of the aquarium that ensures that the minimum amount of glass is used is 12 inch by 12 inch by 6 inch.

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