Chapter 6.4, Problem 60PE

Given a point on the Earth of latitude a, the angle of elevation of the Sun is at its minimum

of $90°-\alpha -23.5°$ at noon at the winter solstice. The angle of elevation of the Sun is at its maximum of $90°-\alpha +23.5°$ at noon on the summer solstice. The angle of elevation follows simple harmonic motion as it varies from its minimum to its maximum over the course of $365$ days. Use this information for Exercises 59-60.

Denver, Colorado, has latitude $39.7°\text{N}$ .

a. Find the angle of elevation of the Sun at noon on the winter solstice and summer solstice.

b. Suppose that the winter solstice is $10$ days before the end of a given year. Write a function representing the angle $A$ (in degrees) of the Sun as a function of the day number $t$ for the following year. (Hint Use a phase shift of- $10$ days.)

c. Graph the function from part (b) on a graphing utility.

d. Determine the angle of elevation of the Sun at noon on January 15 for Denver. Round to the nearest tenth of a degree.