### MODERATE EXERCISES - Modeling with Trigonometric Functions

**(1)** The voltage, V, of an electrical outlet in a home is given as a function of time, t (in seconds), by

_{o}cos(120 t).

- What is the period of the oscillation?
- What does V
_{o}represent? - Sketch the graph of V against t. (Take V
_{o}to be some unknown positive quantity.) Label the axes. - If V
_{o}is 0.5 volts, what is the voltage at time t = 30? - If V
_{o}is 0.5 volts, what is the first time when the voltage is 0.25 volts?

**(2)** A population of animals varies sinusoidally between a low of 700 on January 1 and a high of 900 on July 1.

- Graph the population against time.
- Find a formula for the population as a function of time, t, measured in months since the start of the year.
- According to your formula, what is the population on March 15?

**(3)** A child is swinging on a garden swing with supporting rope lengths of 3.5 m. When the swing angle (with respect to the vertical) is 30^{o}, how high is the child compared with her lowest position?