## Precalculus (6th Edition) Blitzer

$109$ and $231$ days after Jan. 1st.
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.