# What is the smallest time t such that I = 4?

## A generator produces an alternating current according to the equation I = 8 sin 124 $\pi$ t, where t is time in seconds and I is the current in amperes. What is the smallest time t such that I = 4?

Nov 30, 2016

$t \approx 0.0013 \sec o n \mathrm{ds}$

#### Explanation:

$4 = 8 \sin 124 \pi t$

$\frac{4}{8} = \sin 124 \pi t$

${\sin}^{-} 1 \left(\frac{1}{2}\right) = 124 \pi t$

$124 \pi t = \frac{\pi}{6} + 2 \pi n , \mathmr{and} 124 \pi t = \frac{5 \pi}{6} + 2 \pi n$

$t = \frac{\frac{\pi}{6} + 2 \pi n}{124 \pi} \mathmr{and} t = \frac{\frac{5 \pi}{6} + 2 \pi n}{124 \pi}$

$t = \left(\frac{\pi}{6} + 2 \pi n\right) \cdot \frac{1}{124 \pi} \mathmr{and} t = \left(\frac{5 \pi}{6} + 2 \pi n\right) \cdot \frac{1}{124 \pi}$

$t = \frac{1}{744} + \frac{1}{62} n \mathmr{and} t = \frac{5}{744} + \frac{1}{62} n$ where $n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

Since time is positive we are looking for the first positive answer. So pick n values and plug it to the two equations.

$n = 0 , t \approx 0.0013 \mathmr{and} t \approx .00672$

Note that if we pick n = -1 then we get two negative answers and if we pick n = 1 then we get 0.0175 and 0.02285 which are bigger than the values for n = 0 so the smallest time t when I = 4 is about 0.0013 sec.