Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 6060 cm. what is the maximum volume?

Let RR be the radius of the sphere and let hh be the height of the cylinder centered on the center of the sphere. By the Pythagorean theorem, the radius of the cylinder is given by

r2 = R2 - (h2) 2. r2 = R2 - (h2) 2.

The volume of the cylinder is hence

V = π r2 h = π (h R2 - h3 4). V = π r2 h = π (h R2 - h3 4).

Differentiating with respect to hh and equating to 00 to find extrema gives

dV dh = π (R2 - 3 h2 4) = 0 ∴ h0 = 2R 3‾√ dV dh = π (R2 - 3 h2 4) = 0∴ h0 = 2R 3

The second derivative of the volume with respect to hh is negative if h>0 h>0 such that the volume is maximal at h = h0 h = h0. Substituting gives

V max = 4π R3 3 3‾√.