Example 3: Inscribing a Cylinder Into a Sphere

Find the shape of the cylinder of maximum volume which can be inscribed in a given sphere.

The shape of a right circular cylinder can be described by the ratio of the radius of its base to its height. This ratio for the inscribed cylinder of maximum volume should be a number which does not depend on the radius of thesphere. For example, we should get the same shape whether the radius of the sphere is given in inches or centimeters.

Let r be the radius of the given sphere, x the radius of the base of the cylinder, h its height, and V its volume. First, we draw a sketch of the problem in Figure 3.6.4.

Figure 3.6.4

From the sketch we can read off the formulas

V = πx²h, x² + (½h)² = r²,     0 ≤ x ≤ r.

r is a constant. We select x as the independent variable, while h and V are functions of x. To solve the problem we shall find the value of x where V is a maximum and then compute the ratio x/h at this point to describe the shape of the cylinder. The answer x/h should not depend on the constant r. We give two methods of solution.

Solution One: Eliminating One Variable

Sollution Two: Implicit Differentiation