Let $S$ be the surface area of a cube with edge length $L$.  

1. Write $S$ as a function of $L$. 

   Hint (a):

   A cube has six square sides of length $L$. What is the area of each square?

2. If $D$ is the length of the longest diagonal of the cube, write an equation for $D$ as a function of $L$. 

   Step 1(b):

   Draw a triangle within the cube with the longest diagonal $D$.

   Step 2(b):

   $\overline{AC}=\sqrt{L^2+L^2}$

   $\overline{AC}^2=L^2+L^2$

   $\overline{\textit{AC}}=\sqrt{2\textit{L}^2}$

   $\overline{\textit{AC}}=\textit{L}\sqrt{2}$

   Write an equation for the diagonal of the bottom square $AC$.

   Step 3(b):

   Write an equation for the longest diagonal, $D$.

   $D^2=L^2+(L\sqrt2)^2$

   $D^2=L^2+(2L^2)$

   $D^2=3L^2$

   $D=L\sqrt3$

3. Write $D$ as a function of $S$.

   Hint (c):

   Solve for $L$ in your equation from part (a) above. Then substitute that expression for $L$ into your $D$ equation.