Question

A right circular cone has base radius \(4\) cm and height \(10\) cm. A cylinder is to be placed inside the cone with one flat surface resting on the base of the cone. Find the largest possible total surface area (sq. cm) of the cylinder.

• A. \(\frac{100\pi}{3}\)
• B. \(80\pi\)
• C. \(\frac{120\pi}{7}\)
• D. \(\frac{130\pi}{9}\)
• E. \(\frac{110\pi}{7}\)

Hint:

Use similar triangles to relate cone and cylinder dimensions, then optimise with calculus.

Answer 1

The Correct Option is C

Cone’s radius decreases linearly with height. Cylinder radius at height \(h\) from base: \(R(h) = 4\left(1 - \frac{h}{10}\right)\). Cylinder’s surface area: \(S = 2\pi R^2 + 2\pi R h\). Differentiate with respect to \(h\), set to zero for max. Optimal \(h = \frac{10}{3}\) cm, \(R = \frac{8}{3}\) cm. Substitute to find \(S = \frac{120\pi}{7}\).