Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

The answer is 3+ root 69/3.

I really can't think of an approach, yet...

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- Jun 12th 2011, 09:29 PMVeronica1999spheres
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

The answer is 3+ root 69/3.

I really can't think of an approach, yet... - Jun 12th 2011, 09:32 PMProve It
The larger sphere would never be able to rest on two smaller spheres without falling... This question = FAIL!

- Jun 12th 2011, 10:14 PMearboth
1. The centers $\displaystyle (C_1, C_2, C_3)$of the smaller spheres form a isosceles triangle with the side-length 2.

2. The center M of the larger sphere is vertically above the centroid F of the isosceles triangle. The larger sphere is tangent to all three smaller spheres.

3. $\displaystyle MFC_1$ is a right triangle with

hypotenuse $\displaystyle |\overline{MC_1}| = 3$

horizontal leg $\displaystyle |\overline{FC_1}| = \frac 23 \cdot \sqrt{2^2-1^2}=\frac 23 \cdot \sqrt{3}$

vertical leg $\displaystyle |\overline{MF}| =\sqrt{3^2-\left(\frac 23 \cdot \sqrt{3} \right)^2} = \sqrt{\frac{23}{3}}$

4. You have to add one small radius and one large radius to the distance $\displaystyle |\overline{MF}|$ and you'll get the given result.

5. For additional information have a look here: Close-packing of spheres - Wikipedia, the free encyclopedia - Jun 12th 2011, 10:21 PMabhishekkgp
- Jun 12th 2011, 11:34 PMProve It
- Jun 13th 2011, 07:47 AMVeronica1999
Thank you so much Earboth!

- Jun 13th 2011, 01:13 PMbjhopperRe: spheres
also see Earboths solution of a four ball pile 12/22.10

- Jun 13th 2011, 05:08 PMVeronica1999Re: spheres
Thanks for the info. I really appreciate it.