Answer
To produce voltage $E=10\sqrt3$, it takes at least $\frac{10}{3}$ seconds.
Work Step by Step
$$E=20\sin(\frac{\pi t}{4}-\frac{\pi}{2})$$
Since $t$ is time in seconds, the range of $t$ is $[0,\pi)$
For $E=10\sqrt3$, we get the equation
$$20\sin(\frac{\pi t}{4}-\frac{\pi}{2})=10\sqrt3$$
$$\sin(\frac{\pi t}{4}-\frac{\pi}{2})=\frac{10\sqrt3}{20}=\frac{\sqrt3}{2}$$
Over the interval $[0,2\pi)$, $\frac{\pi t}{4}-\frac{\pi}{2}=\frac{\pi}{3}$ is the smallest value which we would have $\sin(\frac{\pi t}{4}-\frac{\pi}{2})=\frac{\sqrt3}{2}$, which would also lead to least positive value of $t$.
$$\frac{\pi t}{4}-\frac{\pi}{2}=\frac{\pi}{3}$$
$$\frac{\pi t}{4}=\frac{5\pi}{6}$$
$$t=\frac{5\pi}{6}\times\frac{4}{\pi}$$
$$t=\frac{10}{3}(seconds)$$
So to produce voltage $E=10\sqrt3$, it takes at least $\frac{10}{3}$ seconds.
