I'm going to bump this, b/c I have to go to class in 4 hours.
EDIT:
NVM, I verified it.
/EDIT
Suppose you make napkin rings by drilling holes with different diameters through two wooden balls( which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure.
(a) Guess which ring has more wood in it.
(b) Check your guess: Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius r through the center of a sphere of radius R and express the answer in terms of h.
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For b, I got that they have the same volume. That was my interpretation after finally getting:
Can anyone tell me if this is correct? If it is, then I don't need any more help, if not, I'll have to post my work.
Watch out, you might get an infraction!
Anyways, there is not enough information in this problem. What does each radius of the cylinders equal?
To compare volumes of WOOD, take the volume of the sphere and subtract the volume of the cylinder from that. This is an estimate because the cylinder gives extra volume that is cut off by the intersection with the sphere.
My expectation is that they didn't give that information, because any sphere of any radius will have the same volume.
I decided this, because I got as my answer, and if all heights are equal, and everything else is a constant, then all volumes must be equal.
But I did a lot of crazy stuff to get that answer, and while I felt confident about all the steps I took, there were many steps involved, and so my confidence in my answer is reduced as there were many oppertunities to make errors.
I figured if this answer was correct, other people on this site would already know it, and would be able to easily say "yeah, they all have the same volume" or "no they don't all have the same volume" in which case I will know if my answer was correct, at least, and could go from there. I guess I thought it would be one of those things that knowledgeable people on this site could just look at and know whether it was correct or not without having to do any calculations.
Okay, I verified that the answer is correct. I thought about it a bit and realized if I simply chose an h and an R, then I could test it.
so I chose h=6, R=9, and got that r=8.49, and V = 113.1
then I chose h=6, R=10, and got that r=9.54, and V = 113.1
So it is correct. I guess I should have looked at that earlier, but I was kind of overwhelmed at the time, studying for midterms, and had a lot of homework in the queue. (Which I got done, w00t).