Answer
$\text{Lowest average monthly temperature }\approx 60.02^o \text{ F}$
$\text{Highest average monthly temperature }\approx 83.98^o \text{ F}$
Work Step by Step
Graph the given function usng a graphing tool to obtain the graph below.
Notice that the average temperature is lowest during the month of January (where $x=1$) and highest during the month of July. (where $x=7$).
Thus,
\begin{align*}
\text{Lowest average monthly temperature} &= 12 \cdot \sin{\left[\frac{\pi}{6}(1-3.9)\right]}+72\\
&\approx 60.02^o \text{F}
\end{align*}
\begin{align*}
\text{Highest average monthly temperature} &= 12 \cdot \sin{\left[\frac{\pi}{6}(7-3.9)\right]}+72\\
&\approx 83.98^o \text{F}
\end{align*}
