Cone, axial angle and inscribed cylinder

Problem

A cone has height $5$ and base radius $1$. a) Find its volume and total surface area. b) A plane perpendicular to the base through the apex $V$ meets the base circle at $A$ and $B$: what is the angle $A\hat V B$? c) Inscribe a cylinder (lower base on the cone’s base, upper base on the lateral surface) with volume equal to $\dfrac{1}{10}$ of the cone’s: find its radius and height.

Solution

a) $V=\dfrac{1}{3}\pi r^{2}h=\dfrac{5\pi}{3}\approx5{,}236$. Slant height $l=\sqrt{r^{2}+h^{2}}=\sqrt{26}$; $S=\pi r^{2}+\pi r l=\pi(1+\sqrt{26})\approx19{,}16$. b) The plane passes through the axis: $A,B$ are endpoints of a diameter, $\overline{AB}=2$ and $\overline{VA}=\overline{VB}=\sqrt{26}$. Then $A\hat V B=2\arctan\dfrac{r}{h}=2\arctan\dfrac15\approx22{,}62^\circ$. c) By similarity, at height $y$ the cone radius is $\rho=1-\dfrac{y}{5}$, so $V_{\text{cyl}}=\pi\rho^{2}y=5\pi\rho^{2}(1-\rho)$. Imposing $V_{\text{cyl}}=\dfrac{V}{10}=\dfrac{\pi}{6}$ gives $\rho^{2}(1-\rho)=\dfrac{1}{30}$, whose solutions are $\rho\approx0{,}205$ ($y\approx3{,}97$) or $\rho\approx0{,}963$ ($y\approx0{,}18$). $\boxed{\,V=\tfrac{5\pi}{3}\approx5{,}24,\quad S=\pi(1+\sqrt{26})\approx19{,}16,\quad A\hat V B\approx22{,}62^\circ\,}$