Consider all right circular cones that have a fixed slant height of 10 cm
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Consider all right circular cones that have a fixed slant height of 10 cm. Each such cone can be described by its base radius r and vertical height h, which are related by the Pythagorean theorem.
(a) Describe how the volume V of the cone varies as a function of its base radius r. Include a discussion of the domain for r in this context.
(b) Outline a calculus‐based procedure for locating the value of r that yields the maximum possible volume under the fixed slant‐length condition. Your answer should state each major step and its purpose without carrying out the detailed algebra.
(c) Explain how you would verify that the critical point obtained indeed corresponds to a maximum volume, citing appropriate theoretical justification.
(d) Interpret, in practical terms, why only one set of dimensions maximizes the cone’s volume given the constant slant height.