Question

the study of alternating electric current requires the solutions of equations of the form ( i = i_{max}sin2pi ft ), for time ( t ) in seconds, where ( i ) is instantaneous current in amperes, ( i_{max} ) is maximum current in amperes and ( f ) is the number of cycles per second. find the least positive value of ( t ), given the following data.
( i = 200, i_{max}=400, f = 50 )
( t=square ) seconds
(round to three decimal places as needed.)

Explanation:

Step1: Substitute the given values into the equation

Given \(i = I_{\text{max}}\sin(2\pi ft)\), substitute \(i = 200\), \(I_{\text{max}}=400\), and \(f = 50\).
We get \(200=400\sin(2\pi\times50t)\).
Then \(\sin(100\pi t)=\frac{200}{400}=\frac{1}{2}\).

Step2: Solve for \(t\) using the inverse - sine function

We know that \(\sin\theta=\frac{1}{2}\) has solutions \(\theta=\frac{\pi}{6}+2k\pi\) or \(\theta=\frac{5\pi}{6}+2k\pi\), \(k\in\mathbb{Z}\).
For \(\theta = 100\pi t\), when \(k = 0\):
If \(100\pi t=\frac{\pi}{6}\), then \(t=\frac{1}{600}\approx0.002\).
If \(100\pi t=\frac{5\pi}{6}\), then \(t=\frac{5}{600}=\frac{1}{120}\approx0.008\).

Answer:

\(t = 0.002\) seconds