第7章 7.26

(1)

体積$V = \dfrac{1}{3}\pi r^{2}h$ $r = \dfrac{h}{3}$ $\dfrac{dV}{dt} = V' = 8$

$V = \frac{1}{3}\pi r^{2}\cdot 3r = \pi r^{3} = 8t \qquad r = 2\cdot\sqrt[3]{\frac{t}{\pi}}$

$V = \frac{1}{3}\pi \frac{h^{2}}{9}\cdot h = \frac{\pi h^{3}}{27} = 8t \qquad h = 6\cdot\sqrt[3]{\frac{t}{\pi}}$

(2)

$h' = 6\frac{1}{3\sqrt[3]{\pi}\sqrt[3]{t^{2}}} = \frac{2}{\sqrt[3]{\pi t^{2}}} \qquad ①$

$h = 9$のとき

$V = \frac{\pi h^{3}}{27} = 27\pi = 8t \qquad t = \frac{27}{8}\pi$

これを①に代入して

$h' = \frac{2}{\sqrt[3]{\pi\cdot\left(\dfrac{27}{8}\pi\right)^{2}}} = \frac{2}{\dfrac{9}{4}\pi} = \frac{8}{9\pi}\ [\text{cm/sec}]$

(3)

$\pi r^{2} = 4\pi\cdot\sqrt[3]{\left(\frac{t}{\pi}\right)^{2}}$

$(\pi r^{2})' = 4\pi\frac{2}{3\cdot\sqrt[3]{\pi^{2}}\sqrt[3]{t}} = \frac{8\pi}{3\cdot\sqrt[3]{\pi^{2}}\,t} = \frac{8\cdot\sqrt[3]{9\pi}}{9}\ [\text{cm}^2/\text{sec}]$