Question

After how many days will there be 12 hours of sunlight for the first time during the year?

Answer 1

In order to have 12 hours of sunlight, the value of the sine will need to be zero, this way S(t) = 12.

The sine can be zero when the argument is 0 or the argument is pi:

$\begin{gathered} \sin(0)=0 \\ \sin(\pi)=0 \end{gathered}$

Since sin(0) is the first day of the year (t = 0), let's use sin(pi):

$\begin{gathered} S(t)=12+2\sin(\frac{2\pi}{365}t)\\ \\ S(t)=12\\ \\ 2\sin(\frac{2\pi}{365}t)=0\\ \\ \frac{2\pi}{365}t=\pi\\ \\ \frac{2}{365}t=1\\ \\ t=\frac{365}{2} \end{gathered}$

Therefore the correct option is the second one.