A right circular cone has height 12 ,cm and base radius 6 ,cm
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A right circular cone has height 12\,cm and base radius 6\,cm. A right circular cylinder is to be inscribed in the cone so that the base of the cylinder lies on the base of the cone and the top of the cylinder touches the interior of the cone.
(a) Let r be the radius of the cylinder (in cm). Express the volume V of the cylinder as a function of r. Clearly indicate the domain of this function that is physically meaningful.
(b) Find the value of r that maximizes V. Provide complete algebraic work leading to all critical points.
(c) Demonstrate that the value of r found in part (b) indeed corresponds to a relative maximum of V by analyzing the sign of V′(r) on intervals surrounding the critical point.
(d) Determine the maximum volume of the cylinder, including proper units, and state the corresponding height of the cylinder.
(e) Explain why no inscribed cylinder can have radius greater than 6\,cm, and describe what happens to the volume function as r approaches that upper bound of the domain.