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 = 2πr

^{2}+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
=> A

_{max}= πr(20-3r) = π(10/3){20-3(10/3)}
=>
A

_{max}= 100π/3**Answer(1)**
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