Chapter 1.3, Problem 198E

Suppose that the number of hours of daylight en a given day in Seattle is modeled by the function $-3.75\cos \left(\frac{\pi t}{6}\right)+12.25$ , with t given in months and t = 0 corresponding to the winter solstice.

a. What is the average number of daylight hours in a year?

b. At which times t₁ and t₂, where $0\le t_1<t_2<12$ , do the number of daylight hours equal the average number?

c. Write an integral that expresses the total number of daylight hours in Seattle between t₁ and t₂.

d. Compute the mean hours of daylight in Seattle between t₁ and t₂, where $0\le t_1<t_2<12$ , and then between t₂ and t₁, and show that the average of the two is equal to the average day length.