第4章 4.15

体積 $V = \dfrac{1}{3}\pi r^2 h$

$r^2 = a^2 - h^2$から

$V = \frac{1}{3}\pi(a^2-h^2)h = \frac{1}{3}\pi(a^2h - h^3)$

$V' = \frac{1}{3}\pi(a^2-3h^2)$

$V'=0$ のとき $\dfrac{1}{3}(a^2-3h^2)=0$ $h^2=\dfrac{1}{3}a^2$

$h = \frac{a}{\sqrt3} = \frac{\sqrt3}{3}a$

このとき

$r^2 = a^2 - \frac{1}{3}a^2 = \frac{2}{3}a^2 \qquad r = \sqrt{\frac{2}{3}}a = \frac{\sqrt6}{3}a$

これより

$V = \frac{1}{3}\pi\left(a^2\cdot\frac{a}{\sqrt3} - \frac{a^3}{3\sqrt3}\right) = \frac{\pi}{3}a^3\left(\frac{3-1}{3\sqrt3}\right) = \frac{2\pi a^3}{9\sqrt3} = \frac{2\sqrt3\pi a^3}{27}$