In the figure shown below, $E$ and $G$ lie on $\overline{AC}$, $D$ and $F$ lie on $\overline{AB}$, $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$, and the given lengths are in feet. What is the length of $\overline{AC}$, in feet?

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$\triangle{ABC}$ is similar to $\triangle{ADE}$. We can set up a proportion:

$$\frac{AB}{AC} = \frac{AD}{AE}$$ $$= \frac{8+7+6}{AC} = \frac{8}{16}$$ $$\frac{21}{AC}=\frac{1}{2}$$ $$AC = \boxed{42}$$