Answer to Question #137121 in Calculus for Kare

Question

A right-circular cone is inscribed in a sphere having a fixed radius of 10 in. Express the volume of the cone as a function of its radius.

Answer 1

The volume of the right circular cone is calculated by the formula.

$V=\frac 1 3\pi r^2H$

where r is the radius and H is the height of the right-circular cone and R=10 sphere radius.

$H=R+\sqrt{\smash[b]{R^2-r^2}}$ or $H=R-\sqrt{\smash[b]{R^2-r^2}}$

Hence the function of the volume of the cone from its base is given as

$V=\frac 1 3\pi r^2H=\frac 1 3\pi r^2×(R±\sqrt{\smash[b]{R^2-r^2}})$

$V=\frac 1 3\pi r^2×(10±\sqrt{\smash[b]{100-r^2}})$ Cubic Inches