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Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height h is h/3.

Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height h is h/3.

Grade:Upto college level

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
9 years ago
look at a side view of the cylinder inscribed in the cone.

in two dimensions it looks like a rectangle inscribed in an isosceles triangle.

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
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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