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.

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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