A right circular cylinder with a base radius of r and height h is inscribed inside a sphere of diameter D such that the outer circular rims of the cylinder rest on the inner surface of the sphere. Which of the following expressions correctly gives height h in terms of D and r?

The correct answer is:
h = $\sqrt{D^2-4r^2}$

Step-by-Step Derivation:

1. Geometry of the sphere and inscribed cylinder:
Consider a right circular cylinder of base radius r and height h inscribed inside a sphere of diameter D.
- The radius of the sphere is R = D / 2.
- The distance from the center of the sphere to the center of either circular base of the cylinder is h / 2.
- The radius of the cylinder's base is r.

2. Applying the Pythagorean Theorem:
A cross-section through the center of the sphere along the cylinder's vertical axis reveals a right-angled triangle formed by:
- The center of the sphere,
- The center of the cylinder's top circular base,
- Any point on the outer circular rim of the top base resting on the inner surface of the sphere.

In this right-angled triangle:

- The hypotenuse is the radius of the sphere, R = D / 2.

- One perpendicular side is half the height of the cylinder, h / 2.

- The other perpendicular side is the base radius of the cylinder, r.

By the Pythagorean theorem:

${\left(\frac{h}{2}\right)}^2+r^2=R^2$

Substitute R = D / 2 into the relation:

${\left(\frac{h}{2}\right)}^2+r^2={\left(\frac{D}{2}\right)}^2$

3. Solving for height h:

$\frac{h^2}{4}+r^2=\frac{D^2}{4}$

Multiply every term by 4 to eliminate fractions:

$h^2+4r^2=D^2$

Subtract $4r^2$ from both sides to isolate $h^2$:

$h^2=D^2-4r^2$

Taking the square root on both sides gives the height h:

$h=\sqrt{D^2-4r^2}$