Answer
$t = 0.050~s$
Work Step by Step
$y(x,t) = 0.040~sin(5\pi x)~cos(40 \pi t)$
We can take the derivative to find an equation for the transverse velocity:
$u = -(0.040)(40\pi)~sin(5\pi x)~sin(40\pi t)$
We can find the values of $t$, where $t \geq 0,$ such that $u = 0$ for all values of $x$:
$u = -(0.040)(40\pi)~sin(5\pi x)~sin(40\pi t) = 0$
$sin(40\pi t) = 0$
$40\pi t = 0, \pi, 2\pi, 3\pi, ...$
$t = 0, 0.025~s, 0.050~s, 0.075~s, ...$
The third time that all points on the string have zero transverse velocity is $~~t = 0.050~s$
