1 sphere on top of 4 spheres

• Apr 7th 2010, 07:47 PM
xiukhung
1 sphere on top of 4 spheres
hi, I have this question.
4 spheres of radius 10 are placed on a table so that each touches 2 others. another sphere is placed on top. Find the height of the top of this fifth sphere above the table.
thanks :)
• Apr 7th 2010, 09:15 PM
Wilmer
• Apr 8th 2010, 01:41 AM
xiukhung
Quote:

Originally Posted by Wilmer

hi, i've read it through but I dont know how this would help. thanks
• Apr 8th 2010, 04:56 AM
Soroban
Hello, xiukhung!

Quote:

Four spheres of radius 10 are placed on a table so that each touches 2 others.
Another sphere is placed on top.
Find the height of the top of this fifth sphere above the table.

The centers of the four spheres form a square with side $\displaystyle 2r.$
It is $\displaystyle r$ units above the table.

The center of the fifth sphere forms a square-base pyramid.
All of its edges have length $\displaystyle 2r.$

We find that the height of this pyramid is $\displaystyle \sqrt{2}\,r.$

The top of the fifth sphere is $\displaystyle r$ units above the pyramid.

Hence, the top of that sphere is: .$\displaystyle r + \sqrt{2}\,r + r \:=\:\left(2 + \sqrt{2}\right)r$ units high.

Answer: .$\displaystyle 10\left(2+\sqrt{2}\right)\text{ units.}$

• Apr 8th 2010, 03:52 PM
xiukhung
A sphere of radius length 8cm rests on top of a hollow inverted cone of height 15 cm whose vertical angle is 60 degrees. Find the height of the centre of the sphere above the vertex of the cone.
• Apr 9th 2010, 01:17 AM
earboth
Quote:

Originally Posted by xiukhung
$\displaystyle \sin(30^\circ) = \frac12 = \frac8x$