An open box, that has a square bottom and rectangular sides, has to have dimensions in order to require the minimum amount of material. The volume of the box is V=256 cube inches. Find the dimensions of the box.
The box has a square bottom and rectangular sides. Let the length of the sides that form the base be represented by x and let the height be y. The volume of the box is 256 cm^3. In terms of x and y, it is x^2*y. As the box is an open one, the area of the material required to construct it is x^2 + 4*xy.
x^2*y = 256
=> y = 256/x^2
substitute this in the expression for the surface area
=> x^2 + 4*x*256/x^2
=> x^2 + 1024/x
To determine the value of x for which x^2 + 1024/x equate its derivative to 0 and solve for x.
=> 2x - 1024/x^2 = 0
=> 2x = 1024/x^2
=> x^3 = 512
=> x = 8
The value of x = 8. y = 256/x^2 = 4
This gives the required dimensions of the box as 8 in. x 8 in. x 4 in.
4inch x 4inch x 16 inch