Answer
In February, the average monthly temperature is $39^\circ F$.
Work Step by Step
$$f(x)=14\sin[\frac{\pi}{6}(x-4)]+50$$
The question asks when the average temperature is $39^\circ F$
In other words, find $x$ so that $f(x)=39$.
Therefore, we can replace $f(x)=39$ to the given equation and find $x$:
$$14\sin[\frac{\pi}{6}(x-4)]+50=39$$
$$14\sin[\frac{\pi}{6}(x-4)]=-11$$
$$\sin[\frac{\pi}{6}(x-4)]=-\frac{11}{14}$$
$$\frac{\pi}{6}(x-4)=\sin^{-1}(-\frac{11}{14})\approx-0.9038$$
$$x-4=(-0.9038)\times\frac{6}{\pi}$$
$$x-4\approx-1.7262$$
$$x\approx2.2738\approx2$$
$x=2$ refers to the month February. Thus, in February, the average monthly temperature is $39^\circ F$.
