Chapter 8.2, Problem 42E

In any given locality, the length of daylight varies during the year. In Des Moines, lowa, the number of minutes of daylight in a day $t$ days after the beginning of a year is given approximately by the formula

$D=720+200\sin \left[\frac{2\pi }{365}\left(t-79.5\right)\right]$, $0\le t\le 365$.

(Source :School Science and Mathematics.)

Graph the function in the window $\left[0,365\right]$ by $\left[-100,940\right]$.

How many minutes of daylight are there on February $14$, that is, when $t=45$?

Use the fact that the value of the sine function ranges from $-1$ to $1$ to find the shortest and longest amounts of daylight during the year.

Use the TRACE feature or the MINIMUM command to estimate the day with the shortest amount of daylight. Find the exact day algebraically by using the fact that $\sin \left(\frac{3\pi }{2}\right)=-1$.

Use the TRACE feature or the MAXIMUM command to estimate the day with the longest amount of daylight. Find the exact day algebraically by using the fact that $\sin \left(\frac{3\pi }{2}\right)=1$.

Find the two days during which the amount of daylight equals the amount of darkness. (These days called equinoxes.) [Note: Answer this question both graphically and algebraically.]