Cylinder in a Cone

Cyliner in Cone There is a cone of radius Rc and height Hc. One has to fit a cylinder in it of radius Rcy. Find the ratio of Rc / Rcy for a) Maximum volume of cylinder & b) For maximum curved surface area of cylinder. Note – Observe the change in volume and surface area with change in height of the cylinder by animating Rcy and one may view cone and different angles by changing α This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com Solution - Let OH be Hcy the height of cylinder. ΔAOC & ΔFHC are similar ∴ CH/CO = Rcy/RC ∴ CH = Rcy*HC/Rc But CH = HC - Hcy ∴ Hcy = HC - Rcy*HC/Rc = HC(1 - Rcy/Rc) Curved surface of cylinder S = 2 π Rcy*Hcy putting value of Hcy = 2 π Hc Rcy(1 - Rcy/Rc) Differentiating S with repect to Rcy and equating to zero 1 - 2Rcy/Rc = 0 Cylinder Curved surface is maximum when Rcy = Rc/2 Height of cylinder is half of height of cone. Volume of cylinder V = π Rcy^2*Hcy putting value of Hcy = π Hcy Rcy^2(1 - Rcy/Rc) Differentiating V wit respect to Rcy and equating to zero 2Rcy - 3(Rcy)^2/Rc = 0 Volume of cylinder is maximum when Rcy = 2Rc/3 height of cylinder is 1/3 of the height of cone. For querries vasantbarve@gmail.com Dr Vasant Barve, 25 August 2014, Created with GeoGebra