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

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

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

$H = R+\sqrt{R^{2}-r^{2}}$

or

$H = R-\sqrt{R^{2}-r^{2}}$

hence the function of the volume of the cone from its base

$V = \frac 1 3 \pi r^{2}H = \frac 1 3 \pi r^{2}*( R+\sqrt{R^{2}-r^{2}}) =\frac 1 3 \pi r^{2}*( 10+\sqrt{100-r^{2}})$

or

$V = \frac 1 3 \pi r^{2}H = \frac 1 3 \pi r^{2}*( R-\sqrt{R^{2}-r^{2}}) =\frac 1 3 \pi r^{2}*( 10-\sqrt{100-r^{2}})$