Chapter 6.4, Problem 59PE

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.

Dallas, Texas, has latitude $32.8°\text{N}$ .

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 February $1$ for Dallas. Round to the nearest tenth of a degree.

e. How many days into the year will the spring and fall equinoxes occur? Round to the nearest day. (Hint: The angle of elevation of the Sun at noon on an equinox will be halfway between its minimum and maximum value. This is the day of the year when the duration of daylight equals the duration of darkness.)