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. Complete the parts below to determine the minimum cost of producing the box.  

1. Sketch a diagram of this situation.

   Hint (a):

   Sketch a box and label the sides of the base $x$ and the height $h$.

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

   Hint (b):

   base cost $=\$3$(area of the base)

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

   Step (c):

   side cost $=\$1$(length of base)(height)
   How many sides does the box have?

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

   Step (d):

   total cost $=$ base cost $+$ cost of all sides

5. Use the fact that the volume of the box is $100$ cubic feet to write an equation for the volume of the box.

   Hint (e):

   $V=$ (area of the base)(height)
   $100=$ (area of the base)(height)

6. Solve for h in your equation from part (e) and then express the cost only in terms of $x$.

7. Graph your equation from part (f) to determine value of $x$ that gives the minimum cost. What is the minimum cost?