Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 25

Use the figure below to find a function that models the number of hours of daylight at
New Orleans as a function of the time of year assuming that the city of New Orleans is
located at latitude $30^{\circ}\rm{N}$. Use the fact that on March 31, the sun rises at 5:51 AM
and sets at 6:18 PM in New Orleans.




Referring to the figure for latitude of New Orleans at $30^{\circ}\rm{N}$, we see that the daylight lasts about 14 hours sometime
in June and 10 hours sometime in December, so the amplitude of the curve is $\displaystyle \frac{1}{2} (14-10) = 2$

Recall that the general equation of sine function is $y = A \sin (2 \pi \rm{ ft})$ or $y = A \sin \displaystyle \left(\frac{2 \pi}{T} t\right)$


$\begin{equation} \begin{aligned} \text{Where } A &= \text{amplitude}\\ f &= \text{frequency}\\ T &= \text{period} \end{aligned} \end{equation}$


Let's say that the curve begins on March 21, the 80th day of the year, therefore the graph of the function
is shifted 80 units to the right and shifted 12 units upward as shown in the figure. Also, we assumed that there are 365 days in a year.
Therefore, the model of the number of hours of daylight at New Orleans as a function of time of year is...

$f(t) = 2 \sin \displaystyle\left[ \frac{2 \pi}{365} (t-80)\right]+12$