Question

What is the largest possible volume of a cone with a slant height of 6?

Answer 1

The volume of a cone is given by:
\[V=\frac{1}{3}\pi r^2h.\]
We have
\(r^2=s^2-h^2 \implies r^2=36-h^2.\) Substituting this into V gives:
\[V=\frac{1}{3}\pi(36-h^2)h=12\pi h-\frac{\pi}{3}h^3 \implies V'=12\pi-\pi h^2.\]
The largest volume occurs at h^2=12 or h=sqrt{12}. Substitute this into V and this would the value of the largest volume.