Question

How do you find the volume of the largest right circular cone that can be inscribed in a sphere of radius r?

Answer 1

The height hh is the distance from the apex to the circular base. It is between 00 and 2r2r (with rr the radius of the sphere), and the radius of the base grows as hh increases from 00 to rr and then shrinks again as hh increases to 2r2r. A diagram may convince you that by Pythagoras (h−r)2+x2=r2(h−r)2+x2=r2, where xx is the radius of the base. That allows you to eliminate x2x2 in favour of 2rh−h22rh−h2, so you have V(h)=13(2rh−h2)hV(h)=13(2rh−h2)h and thus V′(h)=13(4rh−3h2)V′(h)=13(4rh−3h2), which vanishes at h=43rh=43r and thus x=8√3rx=83r, with maximal volume V=13π89r243r=3281πr3=827VsV=13π89r243r=3281πr3=827Vs, where VsVs is the volume of the sphere.

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