Chapter 4.5, Problem 70PE

The duration of daylight and darkness varies during the year due to the angle of the Sun in the sky. The model $d\left(t\right)=2.65\sin \left(0.51t-1.32\right)+12$ approximates the amount of daylight $d\left(t\right)$ (in hours) for Sacramento, California, as a function of the time $t$ (in months) after January $1$ for a recent year; that is, $t=0$ is January $1$ , $t=0$ is February $1$ , and so on. The model $y=n\left(t\right)$ represents the amount of darkness as a function of $t$

a. Describe the relationship between the graphs of the functions and the line $y=12$ .

b. Use the result of part (a) and a transformation of $y=d\left(t\right)$ to write an equation representing $n$ as a function of $t$ .

c. What do the points of intersection of the two graphs represent?

d. What do the relative minima and relative maxima of the graphs represent?

e. What does $T\left(t\right)=d\left(t\right)+n\left(t\right)$ represent?