Q.383. Maximizing area of cylinder inscribed in a sphere.

Question 383.
What is the largest surface area of a right circular cylinder inscribed in a sphere of radius r ?

Answer 383.
Refer to the figure shown below.
Let radius of the cylinder, R = rcosA
and height of the cylinder, H = 2rsinA
=> surface area of the cylinder,
S = 2πR^2 + 2πRH

=> S = 2πr^2 (cos^2 A + 2 sinA cosA)

=> S = πr^2 (1 + cos2A + 2sin2A)

For S to be maximum, dS/dA = 0 and d^2S/dA^2 < 0

dS/dA = 0

=> πr^2 (- 2sin2A + 4cos2A) = 0

=> tan2A = 2 => sin2A = 2/√5 and cos2A = 1/√5

d^2A/dA^2 = πr^2 (-4cos2A - 8sin2A) < 0 for acute A
=> maximum surface area,
S(max)
= πr^2 (1 + 1/√5 + 4/√5)

= πr^2 (1 + √5).

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