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)

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)

Transcript text: 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

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) $.

Step 1: Determine the Amplitude

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

Step 2: Calculate the Angular Frequency

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

Step 3: Formulate the Model for Current

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: \[ I(t) = 4 \cos(376.9911 t) \]

Step 4: Evaluate the Current at $ t = 0 $

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