Answer
In July, the average monthly temperature is $64^\circ F$.
Work Step by Step
$$f(x)=14\sin[\frac{\pi}{6}(x-4)]+50$$
The question asks when the average temperature is $64^\circ F$
In other words, find $x$ so that $f(x)=64$.
Therefore, we can replace $f(x)=64$ to the given equation and find $x$:
$$14\sin[\frac{\pi}{6}(x-4)]+50=64$$
$$14\sin[\frac{\pi}{6}(x-4)]=14$$
$$\sin[\frac{\pi}{6}(x-4)]=1$$
$$\frac{\pi}{6}(x-4)=\sin^{-1}1=\frac{\pi}{2}$$
$$x-4=\frac{\pi}{2}\times\frac{6}{\pi}=3$$
$$x=7$$
$x=7$ refers to the month July. Thus, in July, the average monthly temperature is $64^\circ F$.
