Answer
$6.09$ hours or $7.18$ hours
Work Step by Step
William wants to know the time it will take him to be $50$ miles away from Boise. Substitute $30$ in place of $D(t)$ and solve for $t$. $$\begin{aligned}
D(t) &= \lvert 365-55t\rvert\\
30 &= \lvert 365-55t\rvert.
\end{aligned}$$ Rewrite into two equations and solve. $$\begin{aligned}
365-55t&= 30\\
-55t&=30-365\\
-55t& = -335\\
55t& = 335\\
t&= \frac{335}{55}\\
&= \frac{67}{11}\\
&\approx 6.09
\end{aligned}$$ or $$\begin{aligned}
365-55t&= -30\\
-55t&=-30-365\\
-55t& = -395\\
55t& = 395\\
t&= \frac{395}{55}\\
&= \frac{79}{11}\\
&\approx 7.18.
\end{aligned}$$ Check the solution. $$\begin{aligned}
D(67/11) &= \lvert 365-55\cdot \frac{67}{11}\rvert\\\\
& = \lvert 365-5\cdot 67\rvert\\\\
& = \lvert 365-335\rvert\\
& = \lvert 30\rvert\\
& = 30
\end{aligned}$$ and $$\begin{aligned}
D(79/11) &= \lvert 365-55\cdot \frac{79}{11}\rvert\\\\
& = \lvert 365-5\cdot 79\rvert\\\\
& = \lvert 365-395\rvert\\
& = \lvert -30\rvert\\
& = 30.
\end{aligned}$$ Hence, William will be $30$ miles away from Boise at either $6.09$ hours later of $7.18$ hours later.
