
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 17th, 2012, 04:22 AM  #1 
Joined: Jul 2012 Posts: 3 Thanks: 0  The number of soccer balls to build 4 sided pyramid
How to counts the number of soccer balls it would take to build a 4 sided pyramid (a tetrahedron) of soccer balls 6 foot high. The diameter of each ball is 22cm.

July 17th, 2012, 04:38 AM  #2 
Senior Member Joined: Jan 2010 Posts: 205 Thanks: 0  Re: The number of soccer balls to build 4 sided pyramid
Isn't this related to sphere packing and the Kepler conjecture?

July 17th, 2012, 04:54 AM  #3 
Joined: Jul 2012 Posts: 3 Thanks: 0  Re: The number of soccer balls to build 4 sided pyramid
Probably yes, this problem has connection to sphere packing. But how to approach this problem? I can calculate length of base side of a tetrahedron and divide it by sphere's diameter. From that I know how many spheres will be in the base level and then I can calculate next levels. But is this a good approach? 
July 17th, 2012, 12:19 PM  #4 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,930 Thanks: 319 Math Focus: The calculus  Re: The number of soccer balls to build 4 sided pyramid
Imagine 3 identical spheres of diameter d, all tangent and whose centers lie in the same horizontal plane. Now we place a 4th sphere whose diameter is also d on top so that it is tangent to the other three. If we draw line segments joining the centers of the 4 spheres we form a regular tetrahedron whose side lengths are d. Now, to find the height h of the tetrahedron, we may drop a vertical line segment from the apex to the base and label its length h. Next draw a line segment from the apex down one of its faces bisecting the face, and we find its length from . Now, observing that the vertical line h intersects the base in the middle (the point equidistant from the 3 vertices), we can find the length of the line segment b joining h and l from dividing the equilateral base into 3 congruent isosceles triangles each of which has an area 1/3 that of the equilateral base: Now we have a right triangle whose legs are h and b and whose hypotenuse is l, thus from the Pythagorean theorem, we obtain: From this we may determine how many levels n: We are given and we wish to form a tetrahedral stack high. Hence: Our stack will be just over 6 ft. high. Thus, the total number of balls S is found by adding up the first 10 triangular numbers: For n layers, we would have: 
July 17th, 2012, 01:22 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 11,684 Thanks: 359 
According to the question, d = 22cm, not 22in.

July 17th, 2012, 03:12 PM  #6 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,930 Thanks: 319 Math Focus: The calculus  Re: The number of soccer balls to build 4 sided pyramid
I corrected that and another mistake.


Tags 
balls, build, number, pyramid, sided, soccer 
Search tags for this page 
how many balls in a pyramid stacked 4 high,how to do pyramid with 22cm sides,how many balls can make a four sided pyramid,build pyramid of balls
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Select 5 balls from 12 balls, where some balls are identical  Skyer  Algebra  3  January 6th, 2014 12:26 PM 
Odd/even number of goals in the second half of a soccer game  Mportelli  Advanced Statistics  1  June 24th, 2013 05:11 PM 
Soccer  johnny  New Users  7  May 11th, 2011 05:30 PM 
An urn contains a azure balls and c carmine balls.. (Help)  byron123  Advanced Statistics  2  September 10th, 2008 09:39 AM 
7 balls into 5 boxes, expected number of empty boxes  frendlyfranz  Algebra  8  February 19th, 2008 10:40 AM 