Chapter 4.5, Problem 103PE

Functions a and m approximate the duration of daylight, respectively, for Albany, New York, and Miami, Florida, for a given year for day $t$ . The value $t=1$ represents January $1$ , $t=2$ represents February $1$ , and so on.

$a\left(t\right)=12+3.1\sin \left[\frac{2\pi }{365}\left(t-80\right)\right]m\left(t\right)=12+1.6\sin \left[\frac{2\pi }{365}\left(t-80\right)\right]$

a. Graph the two functions with a graphing utility and comment on the difference between the two graphs.

b. Both functions have a constant term of $12.$ What does this represent graphically and in the context of this problem?

c. What do the factors $3.1$ and $1.6$ represent in the two functions?

d. What is the period of each function?

e. What does the horizontal shift of $80$ units represent in the context of this problem.

f. Use the Intersect feature to approximate the points of intersection.

g. Interpret the meaning of the points of intersection.