Volume of a pencil A pencil sharpened at both ends is formed of a cylindrical barrel together with two conical points. The pencil measures 110mm from tip to tip, has radius $r$ mm, and the cylindrical barrel is $h$ mm long. The lengths of the conical sharpened points are $x$ mm and $y$ mm. Show that the total volume $V$ $mm^3$ of the pencil is given by the formula $$V=(2\pi r^2(55+h))/3$$ and explain why the volume does not depend on the lengths of the sharpened points. All help and solutions are heavily appreciate (been working on this for over an hour and just keep coming up with solutions that do not lead anywhere)! Thank you.

The volume of a cylinder with radius $r$ and height $h$ is $\pi r^2 h$. The volume of a right circular cone with base radius $r$ and height $x$ is $\frac{1}{3} \pi r^2 x$.

Then, the total volume of the pencil is $\pi r^2 h + \frac{1}{3} \pi r^2 x + \frac{1}{3} \pi r^2 y = \pi r^2 ( h + \frac{x + y}{3})$.

Note that $x + y = 110 - h$ and simplify.