Question

a right circular cylinder is inscribed in a right circular cone. Prove that the volume of the largest cylinder is 4/9 of the volume of the cone.
Using Optimisation

Answer 1

Volume of cone: \(\large V=\dfrac{1}{3}\pi R^2H\).

Volume of cylinder: \(\large v=\pi r^2h\).

(Keep in mind that with this notation, smaller letters refer to the dimensions of the smaller solid, and larger to larger.)

We want to maximize \(v\) subject to the fixed constants \(R\), \(H\), and \(V\).