Exercise 55 Page 218

Let's draw a vertical cross-section of the sphere and the cylinder.

• The red segment connects the center of the sphere with a point on the sphere. Its length is the radius of the sphere.
• The vertical segment connects the center of the sphere with the center of the bottom circle of the cylinder. The length is half the height of the cylinder.
• The length of the horizontal leg of the shaded right triangle is the radius of the cylinder.

We can use the Pythagorean Theorem to write a relationship between the height and radius of the cylinder. We can also rearrange this relationship to express the square of the radius in terms of the height.