Home » Blog » Find the largest cylinder that can be inscribed in a cone

Find the largest cylinder that can be inscribed in a cone

by RoRi

Consider a right circular cylinder with radius R and altitude H. Find the right circular cylinder of largest lateral surface area that can be inscribed in this cone.

Rendered by QuickLaTeX.com

The lateral surface area of the cylinder is given by A = 2 \pi r h, where r is the radius of the cylinder and h is the height of the cylinder. From the diagram we find a formula for h in terms of the constants H and R, and the radius of the cylinder r,

    \[ h = - \frac{H}{R}r + H. \]

Thus, letting f(r) denote the lateral surface area we have

    \begin{align*} A = f(r) = 2 \pi r h &= 2 \pi r \left( - \frac{H}{R} r + H \right) \\ &= 2 \pi r H - \frac{2 \pi H}{R} r^2. \end{align*}

Taking the derivative of this with respect to r and setting it equal to zero,

    \[ 2 \pi H - \frac{4 \pi H}{R} r = 0 \quad \implies \quad R - 2r = 0 \quad \implies \quad r = \frac{R}{2}. \]

Since

    \begin{align*} f'(r) &> 0 & \text{when} && r &< \frac{R}{2} \\ f'(r) &< 0 & \text{when} && r &> \frac{R}{2} \end{align*}

Therefore, this critical point is a maximum.
Then plugging this back into our expression for h we have

    \[ h = \frac{1}{2} H. \]

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