Question

if the cylinder of the largest possible volume is inscribed in in a given sphere, the ration of the volume of the sphere to that of the cylinder is what?

Answer 1

*ratio

Answer 2

R - radius of sphere , r - radius of cylinder , h - height of cylinder \[V=\pi r^2h\quad,\quad (2R)^2=(2r)^2+h^2\quad,\quad r^2=R^2-\frac{h^2}{4}\]\[V=\pi\left(R^2h-\frac{h^3}{4}\right)\quad,\quad\frac{dV}{dh}=\pi\left(R^2-\frac{3h^2}{4}\right)=0\]\[\Rightarrow\quad h=\frac{2\sqrt{3}}{3}R\quad\Rightarrow\quad V_{\max}=V=\pi\left(R^2\frac{2\sqrt{3}}{3}R-\frac{1}{4}\left(\frac{2\sqrt{3}}{3}R\right)^3\right)=\]\[=\pi\left(\frac{2\sqrt{3}}{3}R^3-\frac{2\sqrt{3}}{9}R^3\right)=\pi\frac{4\sqrt{3}}{9}R^3\]\[V_S:V_{\max}=\pi\frac{4}{3}R^3:\pi\frac{4\sqrt{3}}{9}R^3=\frac{4}{3}:\frac{4\sqrt{3}}{9}=1:\frac{\sqrt{3}}{3}\]

Answer 3

you are so awesome! thank you!