Problem:A rectangular box is to be open-topped with a volume of . The material for its bottom costs $ , and the material for its four sides costs $ . Find the dimensions of the box that minimize the total cost of the material needed to construct the box.

My Solution:Let = height, = length, = width. Remember that the volume of a rectangular prism is . We can solve for .

Now we can write our equation for the total cost as:

$ $ $

. (1)

We can take the derivative with respect to :

.

so which gives us .

so we can substitute back into (1) and we have:

.

Once again we can take the derivative with respect to :

.

so which gives us and thus or , however must be positive so .

Therefore we have: , , and .

Is that correct? Does that make sense?