Questions: Topic 8. Applications of Trigonometric Functions Solve the problem. An alternating current generator generates current with a frequency of 60 Hz. Suppose that initially, the current is at its maximum of 4 amperes. If the current varies in simple harmonic motion over time, write a model for the current I (in amperes) as a function of the time t (in seconds). Select one: a. I=2 cos (120 pi t)+2 b. I=4 cos (60 t) c. I=2 cos (60 t)+2 d. I=4 cos (120 pi t)

Solution Steps

To model the current as a function of time, we need to use the properties of simple harmonic motion. The general form for simple harmonic motion is \( I(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase shift. Given that the current is at its maximum initially, the phase shift \( \phi \) is 0. The amplitude \( A \) is 4 amperes. The frequency is 60 Hz, so the angular frequency \( \omega \) is \( 2\pi \times 60 \). Therefore, the model for the current is \( I(t) = 4 \cos(120\pi t) \).

The amplitude of the current is given as 4 amperes. This is the maximum value of the current, denoted by \( A = 4 \).

The frequency of the alternating current is 60 Hz. The angular frequency \( \omega \) is calculated using the formula: \[ \omega = 2\pi \times \text{frequency} = 2\pi \times 60 = 376.9911 \]

The current varies in simple harmonic motion, which can be modeled by the equation: \[ I(t) = A \cos(\omega t) \] Substituting the known values, we have: \[ I(t) = 4 \cos(376.9911 t) \]

At \( t = 0 \), the current is: \[ I(0) = 4 \cos(376.9911 \times 0) = 4 \cos(0) = 4 \]

Final Answer