Correct answer: D. 500 * sqrt(2) * pi
Let R be the radius of the sphere, r be the radius of the cylinder, and h be the height of the cylinder. We are given that R = 10. We are also given that the cylinder's height is equal to its diameter, so h = 2r.
To find the relationship between these measures, we can visualize a 2D cross-section of the sphere and the inscribed cylinder. This forms a rectangle (the cylinder's cross-section) inside a circle (the sphere's cross-section). We can form a right triangle where the hypotenuse is the sphere's radius (R), one leg is the cylinder's radius (r), and the other leg is half the cylinder's height (h/2).
By the Pythagorean theorem: r² + (h/2)² = R^2.
Now, substitute the known values and relationships. Since h = 2r, then h/2 = r. Since R = 10, the equation becomes:
r² + r² = 10²
2r² = 100
r² = 50
Now we can find the dimensions of the cylinder. We have r² = 50, so r = √(50) = 5√(2). The height is h = 2r = 2 · 5*√(2) = 10√(2).
The volume of a cylinder is given by the formula V = π · r² · h.
Substitute the values for r² and h:
V = π · (50) · (10√(2))
V = 500√(2) · π
Thus, the correct choice is D.
A is incorrect. This answer results from incorrectly setting up the Pythagorean theorem as r² + h² = R², failing to use half the cylinder's height. This would lead to 5r² = 100, or r² = 20, and a final volume of 80√(5) · π.
B is incorrect. This answer results from correctly finding r² = 50 but then incorrectly using the sphere's radius, R=10, as the cylinder's height in the volume calculation, giving π · 50 · 10 = 500π.
C is incorrect. This answer results from correctly finding r² = 50 but then incorrectly using the sphere's diameter, 2R=20, as the cylinder's height in the volume calculation, giving π · 50 · 20 = 1000π.