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3-48.

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

1. Write an equation for $S$ as a function of $L$

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

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

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

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

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

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

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

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

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 an equation for $D$ as a function of $S$.

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

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