to solve this problem, we need to calculate the volume of the inscribed cylinder.
Let us set
r = radius of cylinder
R = radius of sphere
h = height of cylinder
If we use spherical coordinates and call φ the angle measured from the vertical z axis, the edge of the top face of the cylinder is on the sphere, at a distance R from the center of the sphere (the origin O) and a horizontal distance r from the z-axis. Therefore, using the right triangle:
sin(φ) = r / R
cos(φ) = h / (2R)
Now the volume of the cylinder is simply:
V = Πr2h = 2ΠR3 cos(φ) sin2(φ) = 2ΠR3 cos(φ) ( 1 - cos2(φ) )
which is only a function of the azimuthal angle φ. When φ is zero, h is nearly R and r nearly zero, the cylinder turns into a very thin rod, and then a line. As the angle φ increases, the cylinder gets shorter but thicker.
To find the maximum volume of the cylinder, you can use your calculator and vary the angle φ between 0 degrees and 90 degrees (when the cylinder turns into a very thin, flat pancake).
Mathematically, we can calculate the derivative of the Volume as a function of angle, and equal the derivative to zero:
dV/dφ = 2ΠR3 sin(φ) ( 3 cos2(φ) - 1 )
The 2 solutions which zero out the derivative are φ = 0 (when the sinus is zero), and 3 cos2(φ) = 1
The first solution gives a zero volume (cylinder is a line), the second gives:
cos(φ) = ± 1 / √3, or
φ = 54.73 degrees and 125.26 degrees = 180o - 54.73o. Same cylinder configuration.
The maximum volume is then:
V = 2ΠR3 cos(φ) ( 1 - cos2(φ) ) = 2ΠR3 * 2 / (3√3) = 2.4184 R3
You should approximate this maximum value when using your calculator.
Hope this helps!