Answer
In October, the average monthly temperature in Phoenix is $70.5^\circ F$.
Work Step by Step
$$f(x)=19.5\cos[\frac{\pi}{6}(x-7)]+70.5$$
The question asks when the average monthly temperature in Phoenix is $70.5^\circ F$
In other words, find $x$ so that $f(x)=70.5$.
Therefore, we can replace $f(x)=70.5$ to the given equation and find $x$:
$$19.5\cos[\frac{\pi}{6}(x-7)]+70.5=70.5$$
$$19.5\cos[\frac{\pi}{6}(x-7)]=0$$
$$\cos[\frac{\pi}{6}(x-7)]=0$$
$$\frac{\pi}{6}(x-7)=\cos^{-1}0=\frac{\pi}{2}$$
$$x-7=\frac{\pi}{2}\times\frac{6}{\pi}$$
$$x-7=3$$
$$x=10$$
$x=10$ refers to the month October. Thus, in October, the average monthly temperature in Phoenix is $70.5^\circ F$.
