Anyone know how to find the volume of the largest right circular cylinder...?

Question

Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius r.

~~Calculus Question~~

Answer 1

If you can picture the cylinder inscribed within the sphere, you know that the cylinder intersect the surface of the sphere in two circles, which are the upper and lower bases of the cylinder.

The center of the sphere lies at the center of the cylinder (i.e., the midpoint of the cylinder's vertical axis).

The distance from this point to the sphere is r (of course). But r is also the distance from this point to the circumference of the circles that represent the bases (upper and lower) of the cylinder.

Now, you can think of a line segment from the center of the cylinder to the edge of its base as being the hypotenuse of a right triangle. One leg of the right triangle is the radius of the base, and the other is half of the cylinder's axis.

Conclusion: If you know the length of the cylinder's axis, you can use the above-described right triangle to calculate the radius of
its base(s). And once you know that, you can calculate the cylinder's volume.

Larger conclusion: You can then express the volume of the cylinder as a function of the length of its axis.

Final conclusion: If you take the derivative of the function just mentioned, and set that derivative equal to 0, you can determine the length of axis that will result in the greatest volume for the cylinder.

Two notes:
Of course all these expressions will be in terms of r, since that is the one "given" in the problem.
You might want to check the volume you calculate for the cylinder to confirm that it is less than (but a significant percentage of) (4/3) pi r^3, the volume of the sphere.

Answer 2

Steps:

1. Set up your volume equation (for your cylinder) in terms of R
(sphere) and the angle formed by R and the line perpendicular to h going through the center of the sphere.

2. Differentiate this with respect to theta (the angle).

3. Find the roots to the equation. There will be two roots.

Note: By looking at its graph, you will notice that theta must be between 0 and Pi/2.

Good luck.

Guido

Answer 3

The largest rectangle to fit inside the circle of radius r would be a square of area 2r^2. the integration of this it we rotate the circle around an axis through the centre of the square and perpendicular to one of it's sides would give us the volume of the largest cylinder in a sphere of radius r. Integration would be from 0 to pi R. other wise you'll get double the volume.

Answer 4

Actuator's explanation is very good. Mathematically, I think this comes down to finding the first derivative of:

2(pi)x^2 * sqrt(r^2 - x^2)

The above equation gives the volume of all the cylinders that can be incribed in the sphere. You take the first derivative to get the largest one. Actuator's explanation gives the rationale.