Solve the problem.
-The function $$\mathrm { I } ( \mathrm { t } ) = 40 \sin \left( 60 \pi t - \frac { \pi } { 2 } \right)$$ represents the amperes of current produced by an electric generator as a function of time t, where t is measured in seconds. Find the smallest value of t for which the current is 20 amperes. Round your answer to three decimal
Places, if necessary.
A) 0.008
B) 0.011
C) 0.017
D) 0.033

Question

Quizplus · Accepted Answer

To find the smallest value of $t$ for which the current is 20 amperes, we set the function equal to 20 and solve for $t$:$$20 = 40 \sin\left(60\pi t - \frac{\pi}{2}\right)$$Dividing both sides by 40 gives:$$\frac{1}{2} = \sin\left(60\pi t - \frac{\pi}{2}\right)$$To find $t$, we need to solve for the angle whose sine is $1/2$. The angle $-\frac{\pi}{2}$ shifts the sine wave, but the key is to find where the sine equals $1/2$. The sine of an angle equals $1/2$ at $\frac{\pi}{6}$ and $\frac{5\pi}{6}$ within the first cycle of sine, but due to the shift and the frequency, we focus on the first positive solution after the shift:$$60\pi t - \frac{\pi}{2} = \frac{\pi}{6}$$Solving for $t$ gives:$$60\pi t = \frac{\pi}{6} + \frac{\pi}{2}$$$$60\pi t = \frac{2\pi}{3}$$$$t = \frac{2}{180}$$$$t = \frac{1}{90}$$$$t \approx 0.011$$Thus, the smallest value of $t$ for which the current is 20 amperes, rounded to three decimal places, is approximately 0.011 seconds.

Solve the Problem I(t)=40sin⁡(60πt−π2)\mathrm { I } ( \mathrm { t } ) = 40 \sin \left( 60 \pi t - \frac { \pi } { 2 } \right)I(t)=40sin(60πt−2π​)

Solve the Problem $\mathrm { I } ( \mathrm { t } ) = 40 \sin \left( 60 \pi t - \frac { \pi } { 2 } \right)$