## Friday, 18 November 2011

### Geometry-16 (CAT-2008)

Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest possible total surface area (in sq cm) of the cylinder.
(1)   100 π/3          (2) 80 π/3    (3) 120 π/7    (4) 130 π/9    (5) 110 π/7
Solution follows here:
Solution:
Let the radius and height of the cylinder be ‘r’ and ‘h’ respectively.
Total surface area of cylinder A = 2πr2+2πrh = 2πr(r+h)              ---(1)
As the radius of the inscribed cylinder increases, its height should decrease. At one point the total surface are is maximum. Here A depends on two variables ‘r’ and ‘h’. But to find the maximum value we need to convert the expression into one variable.
GH = height of cone = 10;                DH = base radius of cone = 4;
BE = CF = height of cylinder =h;      CH = base radius of cylinder = r;
CD = DH-CH = 4-r;
Observe that ∆GHD and ∆FCD are similar triangles.
=> CF/GH = CD/DH => h/10 = (4-r)/4 => 2h+5r = 20                   ---(2)
Solving (1) and (2) to eliminate ‘h’,
A = πr(20-3r)
To find maximum value of A, we need to equate its first derivative to zero and solve for ‘r’ value.
A’ = 0 => π{r(-3)+ (20-3r)} = 0 => 20-6r = 0 => r = 10/3
At r=10/3, A has its Maximum value => Amax = πr(20-3r) = π(10/3){20-3(10/3)}
=> Amax = 100π/3