The manufacturing company you work for has been hired to produce a $100$ cubic foot box. It must have a square base and no top. The material for the base costs $$3$ per square foot and the material for the sides costs $$1$ per square foot. We want to write an equation for the cost of the box as a function of $x$, the length of one edge of the base.

1. Write an expression for the cost of the base in terms of $x$.

   Hint (a):

   Write and expression for the area and multiply that by the cost per square foot.

   Hint Answer (a):

   Area: $x^{2}$
   Cost: $3x^{2}$

2. Express the cost of the sides in terms of $x$ and $h$, the height of the box.

   Hint (b):

   Write an expression for the area of one side.
   Write an expression for the area of four sides.
   Write an expression for the cost of four sides.

   Hint Answer (b):

   Area: $4xh$
   Cost: $\left(1\right)\left(4xh\right) = 4xh$

3. Express the total cost of the box in terms of $x$ and $h$.

   Hint (c):

   This is the sum of your expressions from parts (a) and (b).

4. Use the fact that the volume of the box is $100$ cubic feet to eliminate h from the equation you just wrote and express the cost solely in terms of $x$.

   Step 1(d):

   $100 = x^{2}h$

   $h = \frac{100}{x^2}$

   Step 2(d):

   Substitute 'h' into the equation in part (c).

5. Graph the equation and find the value of $x$ which gives the minimum cost. What is the minimum cost?

   Graph (e):