
$V = \frac{1}{3} \pi r^2 h$
Where:
$h = s \sin \theta$
$r = s \cos \theta$
$V = \frac{1}{3} \pi s^3 \cos^2 \theta \sin \theta$
$V = \frac{1}{3} \pi s^3 (1 - \sin^2 \theta) \sin \theta$
$V = \frac{1}{3} \pi s^3 (\sin \theta - \sin^3 \theta)$
$\dfrac{dV}{d\theta} = \frac{1}{3}\pi s^3 (\cos \theta - 3\sin^2 \theta \cos \theta) = 0$
$\cos \theta - 3\sin^2 \theta \cos \theta = 0$
$1 - 3\sin^2 \theta = 0$
$\sin^2 \theta = 1/3$
$\sin \theta = \sqrt{1/3} = 1 / \sqrt{3}$
$h = s \sin \theta = \frac{1}{\sqrt{3}}\,s$
$r = s \cos \theta = \sqrt{\frac{2}{3}}\,s$ answer