Question

HELPPPPP QUICKLYYY !!! The picture shows a cylinder of height x which has been inscribed in a sphere of radius r. Find the volume of a cylinder in terms of x.

Answer 1

See attached. We can draw a right triangles contained within both shapes.

The triangle has height x (same as the cylinder), base 2r (same as the diameter of the cylinder), and hypotenuse 6 (same as the diameter of the sphere).

Since it's a right triangle, the Pythagorean theorem holds and we have

x² + (2r)² = 6²

which we solve for r to get

x² + 4r² = 36

4r² = 36 - x²

r² = 9 - x²/4

r = √(9 - x²/4)

Then the volume of the cylinder, whose height is x and whose base has radius r, is

π r² x = π (9 - x²/4) x = (36x - x³) π/4