Cylinder in a cone A right circular cylinder is placed inside a cone of ra ? dius? and height ? H so that the base of the cylinder lies on the base of the cone. a. Find the dimensions of the cylinder with maximum volume. Specifically, show that the volume of the maximum-volume cylinder is the volume of the cone. b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).
Solution 47E Step 1: (a) Consider the base radius and height of the cone are R and H Let r and hBe the base radius and height of the cylinder Step 2 : Graph of cylinder in a cone is shown below The sides of the similar triangle are in proportion R r H = Hh Solving for h gives Hh = H r R Hr H h = R h = H HR Now the volume of the cylinder is 2 V = r h Hr = r (H R) V = r H Hr3 R Step 3 In order to minimize the volume of cylinder we need to differentiate v with respect to r and equate it to 0 That isdV = 0 dr d Hr3 dr( r H R )=0 2rH 3r 2H = 0 R rH(2 ) R 0 2R3r rH( R ) = 0 This gives r=0 which obviously gives the minimum volume of the cylinderV = 0 2R r = 3 gives maximum volume Step 4 : Now we have h = H Hr R 2H 2R h = H 3 (since r = 3 ) h = H 3 2R H So the dimensions of the cylinder are r = 3 and h = 3 Step 5 And the maximum volume of the cylinder is 3 V = r H Hr R 2R 2 H 2R 3 V = ( )3H R ( 3 4R H 8HR2 = 9 27 2 2 = 12R H8HR 27 2 = 427R