Question

1. Calculate the height of a cylinder of maximum volume that can be cut from a cone of height 20 cm and base radius 80 cm.

Answer 1

Let the cylinder have height h and radius r. Using similar triangles,

(20-h)/r = 20/80 = 1/4
h = 20 - r/4
v = 4πr^2 h = 4πr^2 (20-r/4) = 80πr^2 - πr^3
dv/dr = 160πr - 3πr^2 = πr(160=3r)
r = 160/3
In general, maximum volume is when r = 2/3 R (radius of cone)
so now use that to find v