Assume for this question that each month has 30 days and that over the course of one year the number of hours of daylight per day in Fairbanks, Alaska is a sinusoidal function of time. Suppose also that Fairbanks gets a maximum of 21.5 hours of daylight/day on June 21 and a minimum of 2.5 hours of daylight/day on December 21.

(a) Define a function that expresses the number of daylight hours/day in terms of the number of days since June 21. Then sketch by hand a graph of the function over the course of one year. Be sure to label the axes with the appropriate variable names and units.

(b) Use the graph from part (a) to find an expression of your function. State an appropriate domain that models one year. Explain your reasoning.

(c) Set up and evaluate a definite integral that gives the total number of day- light hours in Fairbanks over the course of the summer. Then use technology to approximate this total. Explain your reasoning.

(d) Compute the average number of daylight hours per day over the course of a summer in Fairbanks. Explain the logic behind this computation.

(e) What fraction of the summer has passed on the day that the cumulative number of summer daylight hours reaches half of the summer’s total? Answer this question algebraically without technology. You should start by defining an unknown.