Answer
$\frac{1}{8}$ seconds
Work Step by Step
To find the time when the weight first reaches maximum hieght, we need to substitute the maximum hieght (amplitude) into the formula and then solving for $t$:
$s(t)=-4\cos 8\pi t$
$4=-4\cos 8\pi t$
$\frac{4}{-4}=\cos 8\pi t$
$-1=\cos 8\pi t$
$\cos^{-1} -1=8\pi t$
$\pi=8\pi t$
$8\pi t=\pi$
$t=\frac{\pi}{8\pi}=\frac{1}{8}$ seconds
Therefore, the weight first reaches maximum height after $\frac{1}{8}$ seconds.