Oren S. answered • 04/18/21
BA in Mathematics with 5 Years of Teaching Experience
What an interesting question! We will begin by visualizing the shape. To achieve maximum volume, the edges of the cylinder (the circles around the two flat faces) will be on the surface of the sphere, with the rest of the cylinder in the interior of the sphere. You can imagine two such cylinders; one where the cylinder is long and narrow, almost just a diameter of the sphere, and one where the cylinder is wide and flat, close to a circle bisecting the sphere. The correct answer will be in between these extremes.
The volume of the cylinder is πr2h, where r and h are the radius and height of the cylinder. Due to the restriction that the cylinder must be inscribed within the sphere, we can solve for r2 in terms of h. We know that the distance from the center of the cylinder to any point on either edge is 15 cm. Thus by the Pythagorean Theorem, r2 + (h/2)2 = 152. Algebraic manipulation gives us r2 = 225 - h2/4. Plugging this into our original expression for the volume of the cylinder gives us πh(225 - h2/4) or π(225h - h3/4).
Since this is a maximization problem, we simply take the derivative of the expression we reached in the previous paragraph, set it equal to 0, and solve for h. We have π(225 - 3h2/4) = 0, so h = √300 or 10√3. Plugging that into our earlier formulas, r2 = 225 - √3002/4 so r = √150 = 5√6 and V = π√1502√300 = 1500π√3.
We've solved the problem, but let's check to see if our answer makes sense before we finish. The height is about 17 cm and the radius is about 12 cm, which seem reasonable. The volume of the sphere is 4500π, so the sphere is √3 times as large as the cylinder. Why do I insist on doing these extra calculations? Because when I originally solved this problem, I thought I was done, but then I calculated the cylinder was bigger than the sphere so I knew I had to go back and fix a mistake. Now the answer does not violate common sense.
I hope this answer was helpful to you!
