# Thread: 1 sphere on top of 4 spheres

1. ## 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

2. Originally Posted by Wilmer
hi, i've read it through but I dont know how this would help. thanks

3. Hello, xiukhung!

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

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.

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