A paper cone is to be formed by starting with a disk of radius 9cm, cutting out a circular sector, and gluing the new edges together. The size of the circular sector is chosen to maximize the volume of the resulting cone. How tall is the cone?

Answer 1

consider the sector subtended by an angle θ. The arc length to be removed is thus 9θ, which will be the circumference of the base of the cone. The base radius of the cone is thus

r = (18π - 9θ)/2π
Looking at a cross-section of the cone, we see that r^2 + h^2 = 9^2

The volume of the cone is

v = 1/3 π r^2 h
= 1/3 π r^2 √(81-r^2)

dv/dr = πr(54-r^2)/√(81-r^2)
dv/dr=0 when r^2 = 54, so r = 3√6

so, h^2 = 81-54 = 27
h = 3√3

Hmm. I wonder if h=R^(3/2) if the original circle is of radius R?