A right circular cylinder is inscribed in a sphere of radius r.?

Question

Find the largest possible surface area of such a cylinder.

Surface area of cylinder = 2*pi*r^2 + 2*pi*r*h

(Note: This is an optimization problem involving maximum values.)

Answer 1

Assume that the sphere is centered at (0,0,0)
Start out by slicing the sphere through its center and looking at the cross section of the sphere and the cylinder. You will have a rectangle inscribed inside a circle.


Draw a line from the center of the rectangle to the corner of the rectangle that is in the first quadrant. It will have length r since it is a radius of the sphere. Let t be the angle the radius makes with the positive x axis.

The height of the cylinder is r*sin(t).
The radius of the cylinder is r*cos(t)

The surface area is 2*pi*r^2cos^2(t) + 2*pi*rcos(t)rsin(t)

Take the derivative and find the max.

Answer 2

A = 2πr^2 + 2πrh
r = Rsinθ, h = 2Rcosθ
A = 2πR^2(sin^2θ + 2sinθcosθ)
dA/dθ = 4πR^2(sinθcosθ - sin^2θ + cos^2θ)
Maximizing,
sinθcosθ - sin^2θ + cos^2θ = 0
tan^2θ - tanθ - 1 = 0
tanθ = (1 ± √5)/2 = -0.618, 1.618
θ = 58.2825
A = 2πR^2(sin^θ + 2sinθcosθ)
A = 5.61985R^2