Answer
$t=\frac{k}{2}$ where $k=1,2,3,...$
Work Step by Step
Step 1. Given the equation $y=4e^{-3t}sin2\pi t$, since $e^{-3t}\gt0$, $y=0$ requires $sin2\pi t=0$
Step 2. The general solutions for the equation $sin2\pi t=0$ are $2\pi t=k\pi$ where k is a positive integer. Thus
$t=\frac{k}{2}$
Step 3. Exclude $t=0$, the first zero happens when k=1 and $t=\frac{1}{2}$
