Mean Radius from midpoint of cylinder to boundry surface?

Question

That is the question, the answer, if you are using a radius of 10 and a height of 20 is 11.32...

THAT IS THE CORRECT ANSWER.

Please do not tell me how to solve this if you can't get that answer.

I will give the points right away, thank you in advance for any help you can provide

Let me be a little more explicit, I am trying to calculate the mean radius FROM the midpoint of a cylinder whose radius is 10 and height is 20, TO its boundary surface.

The answer is 11.32.

Let me be even more explicit...

If this were a sphere, it would be calculating using

mean radius = 1/(4pi)*Integral(theta=0 to theta=pi)*Integral(Beta=0 to Beta=2pi)*rsin(theta)*dthetha*dbeta

Answer 1

The use of the term "mean" here is pretty odd. You are calculating the radius to points on the cylinder but you are not weighting the values based on the cylinder but instead based on the differential area of a spherical surface. I only discovered this by implementing weird interpretations of the term "mean" until I got the coveted "right" answer. I see no physical reason to use a spherical surface in your weighting although for some specific problem it could be valid. In that case you should state it instead of using the nebulous "mean". Otherwise, you could choose a cube or an ellipsoid for your integration surface and get different but equally valid answers.

That aside, here is how you get your answer:

You need to use a spherical surface differential element (r^2 * cos(phi)* dphi * dtheta with phi as the latitude, not colatitude, and theta as the azimuth) and a definition of the radius in spherical coordinates for the walls (r/cos(phi)) and lid (h/(2 * sin(phi))) separately. The resulting integrals are pretty simple. The two integrations are simply added to produce the weighted sphere area-cylinder radius product. Since the whole thing is weighted on the spherical area, you need to divide by the sphere area to get the mean.

The closed form relation for any such spherically-weighted-mean cylinder radius is:

Define an aspect ratio: k = h/(2*R)

Mean Radius = R*(atan(k) + (k/2) *Ln(1 + 1/(k^2)))

Answer 2

IS this a fluid dynamics problem ?