Answer
$ 109$ and $ 231$ days after Jan. 1st.
Work Step by Step
Step 1. Given the model function
$y=3\ sin[\frac{2\pi}{365}(x-79)]+12$
at
$y=13.5\ hours$, we have
$3\ sin[\frac{2\pi}{365}(x-79)]+12=13.5$
or
$sin[\frac{2\pi}{365}(x-79)]=0.5$
Step 2. The general solutions for the above equation are
$\frac{2\pi}{365}(x-79)=2k\pi+\frac{\pi}{6}$
and
$\frac{2\pi}{365}(x-79)=2k\pi+\frac{5\pi}{6}$
or
$x=365k+\frac{365}{12}+79$
and
$x=365k+\frac{1825}{12}+79$
where $k$ is an integer.
Step 3. Within a year of $x\in[0,365)$ days, we have $x\approx109$ and $x\approx231$ days after Jan. 1st.
