### ADVANCED EXERCISES - Modeling with Trigonometric Functions

**(1)** A patient in the hospital had an illness in which his temperature (in degrees Celsius) varied from a low of 37^{o} to a high of 40.4^{o}. The length of time between successive highs is 16 days.

- Determine the formula for the temperature, T, of the patient at time t in days since the beginning of the illness. Assume that the function describing the temperature can be modeled with a sine function, with no phase shift.
- Sketch a graph of the function over the interval 0 t 20.
- According to this model, what was the patient's original temperature?
- What is the patient's temperature on day 4 of his illness?
- When does the first temperature high occur?

**(2)** The Bay of Fundy is reported to have the largest tides in the world, with the difference between low and high level being as much as 15 meters. Suppose at a particular point in the Bay of Fundy, the depth of the water, y, in meters, as a function of time, t, in hours, from midnight on January 1, 1994, is given by

_{0}+ a cos[b(t - t

_{0})].

- What is the physical meaning of y
_{0}? - What is the value of a?
- What is the value of b? Assume the time between successive high tides is 12.5 hours.
- What is the physical meaning of t
_{0}?