Answer
$53.21^{\circ}\text {F}$
Work Step by Step
The average monthly temperature for the month of December can be found by substituting $x=12$ in the formula:
$y=67.5+16.5\sin [\frac{\pi}{6}(x-4)]\\$
$y=67.5+16.5\sin [\frac{\pi}{6}(12-4)]\\$
$y=67.5+16.5\sin [\frac{\pi}{6}(8)]\\$
$y=67.5+16.5\sin [\frac{4\pi}{3}]\\$
$y=67.5+16.5(-0.8660254038)\\$
$y=67.5-14.28941916\\$
$y=53.21058084\approx53.21$
Therefore, the temperature in December is approximately $53.21^{\circ}\text{ F}$
Using the graph and locating the point when $x=12$ gives $53.211^o\text{ F}$ (refer to the graph below), which matches the answer above.
