[Guest]

First puzzle on here.....I saw this recently and thought I'd post it here for all you math-minded individuals.

There was an answer provided (which I have verified by drawing it out), but this wasn't too satisfying and didn't give a proof!

So...

I have 3 circles (or cylinder, pipes, cigars if you like). Two lie just touching each other (with radii of 9 and 12cm), with their bases on the ground while the third rests on top so it touches both of them and its top is at the same height as the top of the largest circle.

What is the maximum diameter of the little circle at the top? Can you prove it?

Enjoy....

[Guest]

If you join all centers of all three circles, you should form right angle triangle at the center of smaller circle.

So (x/2 + 4.5)² + (x/2 + 6)² = (4.5 + 6)²

Solve the equation for x will give answer. 4.2733 cm.

[Morningstar]

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Edited November 30, 2011 by Morningstar

[Morningstar]

What? Why did my post just break? I suppose it doesn't matter. But I wonder what "<blank>" means.

[Guest]

esridhar - close but not right

My original approach was the same (unfortunately I don't have my workings with me else I would post them).....but something, I forget what, didn't add up when assuming right-angled triangle. If however, you can prove that the angle in the small circle is 90 degrees, I will be more than happy to accept your answer over the published one. The actual answer is very neat, but is stated as a property of stacked cylinders....and doesn't say how that is arrived at!

So I'm hoping some of the brainboxes on here will arrive at the right answer and show how it works...

If nobody gets the answer in a while, I'll post it and perhaps we can reverse engineer it

[curr3nt]

The x I have here is the radius not the diameter.

(6-x)² + y² = (6+x)²

y = √24x

(7.5-x)² + z² = (4.5+x)²

z = √(24x - 36)

Which makes...

(y + z)² + 1.5² = 10.5²

Edited November 30, 2011 by curr3nt

[plasmid]

I don't think the angle in the small circle formed by connecting the centers of the circles can be assumed to be 90 degrees. Consider if you take the 9 cm circle and gradually shrink it until it's just 6 cm ... then the smaller circle would also be 6 cm in order to reach the 12 cm line, and those 6 cm circles would be sitting on top of each other, just touching the 12 cm circle, and the angle inside the top circle would be much less than 90 degrees.

For this problem, I would have taken the approach of saying there are three variables (radius, x, and y coordinates of the small circle) with three equations (distance from the 9 cm circle, distance from the 12 cm circle, and distance from the top line at 12 cm) and tried to solve that system of equations. But it sounds like there's some theorem or something that makes this solvable much more easily, and I don't know of such a theorem.

[Guest]

The diameter = 4cm. You can't assume a right triangle in the center to center triangle, however you can find the answer using a system of right triangles. Assuming the largest is cylinder A, middle is cylinder B, and unknown is cylinder C.

First right triangle: Center of A to Center of B, can solve for horizontal distance v between the center of B to the vertical line of cylinder A (v= sqrt(10.5^2 - 1.5^2)_

Second right triangle: Using center of B to center of C, we can find T, the horizontal distance between C and the vertical line of B. (T= sqrt((4.5+r)^2 - (7.5-r)^2 , T= sqrt(24r-36))

Third: Using center of A to C (6+r), we can find the horizontal distance u of cylinder C to vertical line of A ((6+r)^2 = (6-r)^2 + u^2.

Then, knowing that u+t=v, we can solve for r, giving us 2, which is a diameter of 4.

Edited November 30, 2011 by dummy007

[curr3nt]

I get 4 as the diameter.

[Guest]

Yep. I also get 2 for the radius and thus 4 exactly as the diameter.

[Guest]

<p>Yep, 4 it is! The solution given says that when cylinders are stacked in this way, the product of the diameters for the two smaller circles is equal to the square of the radius for the larger circle => so x*9 = 6^2 --> x=4<br></p><p></p>

Well done dummy007, curr3nt and AntonMagus for the correct answer.......and my tired brain cells also thank dummy for giving such a nice little proof, 'tis much better than the one given with the question

[Guest]

(x+9)^2+(x+12)^2=21^2

x^2+21x+108=441

x^2+21x-333=0

(x+10.5)(x+10.5)=333-110.25

(x+10.5)^2=222.75

x=10.5+(22.75)^0.5

x=10.5-(22.75)^0.5

As it is a length (rather than a vector), take the positive length: x=10.5+(22.75)^0.5

Therefore, x=4.769696007084728

x=4.77 to two decimal places

[Guest]

Sorry David but nope - I think you might have the diameter confused with radius in the equation (plus, can't assume right-angled triangle )....

post #8 has the answer if you're interested.

*** and remember spoilers if you think you have a solution ***

[Guest]

3-ish cm