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.
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:
$ $ $
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?