Question

A right circular cylinder is inscribed in a cone of height h and radius r. Determine the dimension of the cylinder with the largest possible volume.

Answer 1

Draw a diagram of the cross-section.
If the height and radius of the cylinder are h and r respectively,
then from similar triangles (or taking tan of the semi-vertical angle
r/(H-h)=R/H so h=H(1 - r/R)
The vol of the cylinder V=pir^2h=pir^2H(1-r/R)=piH(r^2 -r^3/R)
This is max when r^2-r^3/R is max which is when 2r-3r^2/R =0 (dV/dr=0)
giving r=2R/3 and h=H/3

Answer 2

So would the cross section of a cylinder be a rectangle?

Answer 3

i need go sorry

Answer 4

ok thanks anyways