# Math Help - daylight

1. ## daylight

in a given region, the number of daylight hours vaires, depending on the time of year. this variation can be approximated with a sinusoidal function. the model for a certain region is given by the function d(t)= 5sin 2pie/365 (t-95)+13, where d(t) is in hours and t represents the number of days after January 1. find two days whent he approximate number of daylight hours is 16h.

2. Originally Posted by anitha
in a given region, the number of daylight hours vaires, depending on the time of year. this variation can be approximated with a sinusoidal function. the model for a certain region is given by the function d(t)= 5sin 2pie/365 (t-95)+13, where d(t) is in hours and t represents the number of days after January 1. find two days whent he approximate number of daylight hours is 16h.
$d(t)=5\sin\left(\frac{2\pi(t-95)}{365}\right)+13$

The number of daylight hours you want is 16, so let d = 16...

$16=5\sin\left(\frac{2\pi(t-95)}{365}\right)+13$
$3=5\sin\left(\frac{2\pi(t-95)}{365}\right)$
$\frac{3}{5}=\sin\left(\frac{2\pi(t-95)}{365}\right)$
$\sin^{-1}(\frac{3}{5})=\frac{2\pi(t-95)}{365}$

The value of $\theta$ for which $\sin \theta = \frac{3}{5}$ is $0.643501$ and $\pi-0.643501=2.49809$

$\{0.643501,2.49809\}=\frac{2\pi(t-95)}{365}$
$\frac{365}{2\pi}\times\{0.643501,2.49809\}=t-95$
$\{37.382,145.118\}=t-95$
$\{132.382,240.118\}=t$

So the amount of time is about 132 and 240 days after January 1 (respectively).

Or the days are May 12 and August 28.