An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material requi

Question

An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. What dimensions minimize the amount of material required to build this tank? Show that your result is a minimum.

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Ella 4 days 2021-10-09T18:59:46+00:00 1 Answer 0

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    2021-10-09T19:01:23+00:00

    Answer:

    x  =  8  ft

    h =  1/2  ft  

    Step-by-step explanation:

    Let  x be side of the base then area of the base is  x²

    Let h be the height of the tank    

    Tank volume is   32 ft³     and is   32  =  x²*h    then  h  = 32 /x²

    Area of base  + lateral area = total area (A)

    A = x²  + 4*x*h      ⇒   A = x²  + 4*x*(32/x²)          A = x²  + 128/x

    A(x)  = x²  + 128/x     (1)

    Taking derivatives on both sides of the equation

    A´(x)   =  2x   –  128/x²             A´(x)   =  0             2x   –  128/x²   =  0

    (2x² -128) / x²    =  0

    2x²  –  128  = 0

    x²  =√64

    x  =  8  ft

    The result is minimum since replacing in equation (1)  x = 8 we get

    A(x) > 0

    And

    h  =  32/x²

    h =  1/2  ft  

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