Math CentralQuandaries & Queries


Question from astrogirl, a student:

find the volume of the biggest right circular cone that can be inscribed in a sphere of radius a=3


In my diagram the sphere has centre C and radius R. The cone has base radius r and height h.

sphere and cone

The volume of a right circular cone is V = 1/3 π r2 h. To apply the calculus you know you need to express this volume as a function of one variable. The right triangle ABC give the information you need.

By Pythagoras theorem

|CA|2 + |AB|2 = |BC2|


(h - R)2 + r2 = R2


r2 = R2 - (h - R)2

Substitute r2 into the expression for the volume of the cone and you have then a function of one variable h. Use your calculus to find the value of h that maximized the volume.


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