Doug C. answered • 05/30/18

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Hi Brandon,

Here is a start for you. You might want to draw a rough sketch of the box (without a top).

Label the width of the bottom x. The length of the base is represented by 4x. For now the height of the box is h. We want to minimize surface area, but in terms of one variable. The reason you are given the volume of the box is so that you can establish a relationship between x and h.

V = lwh.

For your purposes that will be 100 = 4x(x)(h). Therefore h = 100/4x

^{2}= 25/x^{2}.Now you need a formula for the surface area. The base has an area of 4x(x). Two of the sides of the box have an area defined as xh. The other two sides have an area 4x(h).

S = 4x

^{2}+ (perimeter of the base)h = 4x^{2}+ 10x(h) = 4x^{2 }+ 10x(25/x^{2}) = 4x^{2}+ 250x^{-1}Now take the derivative of S with respect to x. Set that equal to zero to find critical numbers. Use some kind of test to convince yourself that the critical number generates a minimum surface area. Use the value of x to determine the other two dimensions (4x and 25/x

^{2})