S. Show that the height of a right circular cylinder of maximum volume

1.	S. Show that the height of a right circular cylinder of maximum volume that can beinscribed in a given right circular cone of height h is
Answer» let the base of this triangle lie on the x-axis, with the two sides of the triangle symmetrical to the y-axis. let the base of the triangle = 2R, height = H side of the triangle in quad I has equation y = H - (H/R)x base of the cylinder = x height of the cylinder = y = H - (H/R)x so, cylinder volume is ... V =x[H - (H/R)x] = H[x- x/R] dV/dx = H[2x - 3x/R] set dV/dx = 0, x(2 - 3x/R) = 0 x = 2R/3 ... the cylinder radius for max volume so height of the cylinder=y=H-(H/R)(2R/3) y=H/3

S. Show that the height of a right circular cylinder of maximum volume that can beinscribed in a given right circular cone of height h is

Answer»

let the base of this triangle lie on the x-axis, with the two sides of the triangle symmetrical to the y-axis.

let the base of the triangle = 2R, height = H

side of the triangle in quad I has equation y = H - (H/R)x

base of the cylinder = x

height of the cylinder = y = H - (H/R)x

so, cylinder volume is ...

V =*x*[H - (H/R)x] = H[x- x/R]

dV/dx = H[2x - 3x/R]

set dV/dx = 0, x(2 - 3x/R) = 0

x = 2R/3 ... the cylinder radius for max volume

so height of the cylinder=y=H-(H/R)(2R/3)

y=H/3