Home » Blog » Find the largest cylinder (in terms of volume) that can be inscribed in a given cone

Find the largest cylinder (in terms of volume) that can be inscribed in a given cone

by RoRi

Given a right circular cone with radius R and height H, find the right circular cylinder of maximum volume that can be inscribed in the cone.

Rendered by QuickLaTeX.com

The setup is the same as in the previous exercise, except this time we want to maximize the volume V = \pi r^2 h instead of the lateral surface area. Again, we have the following expression for h,

    \[ h = \left( \frac{-H}{R} r + H \right). \]

Then,

    \[ V = \pi r^2 h = \pi r^2 \left( -\frac{H}{R} r + H \right) = \pi H r^2 - \frac{\pi H}{R}r^3. \]

Taking the derivative of this with respect to r,

    \[ \frac{dV}{dr} = 2 \pi H r - \frac{3 \pi H}{R} r^2. \]

Setting this equal to 0 we have,

    \begin{align*} \frac{dV}{dr} = 0 && \implies && 2 \pi H r - \frac{3 \pi H}{R} r^2 &= 0 \\ && \implies && 2 \pi H - \frac{ 3 \pi H}{R} r &= 0 &(r \neq 0) \\ && \implies && 1 - \frac{3}{2R} r &= 0 \\ && \implies && r = \frac{2R}{3}. \end{align*}

Plugging this into our expression for h we then have

    \[ h = \frac{H}{3}. \]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):