Abox, with rectangular sides, base and top is to have a volume of 4 cubic feet. it has a square base. if the material for the base and top costs 10 dollars per square foot and that for the sides costs 20 dollars per square foot, what is the least cost it can be made for?

C(min) = 120 $

Step-by-step explanation:

Let x be the side of the square Then

Area of the base b equal to area of the top

A(b)  =  A(t)   =  x²

Cost (base + top )     C₁ =   2 * 10 * x²    C₁ =  20*x²

Sides  area    are equal

x*h     by   4             and      V of the box  is     V  =  4  =  x²*h  

then     h  =  4/x²

Area of all sides = 4* x* 4/x²        16/x

And cost of sides area is

C₂  =  20*16/x

Total cost ($)

C   =  C₁   +  C₂

C(x)   =  20 x²  +  320/x

Taking derivatives  on both sides of the equation

C´(x)   =  40 *x  - 320/x²

C´(x)  =  0         40 *x  - 320/x² = 0    x³  - 8  = 0

x = 2

And minimun cost

C(min) =  20 * (2) + 320/(2)²

C(min) = 40  + 80  

C(min) = 120 $

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