Billiard balls

A layer of ivory billiard balls of radius 6.35 cm is in the form of a square. The balls are arranged so that each ball is tangent to every one adjacent to it. In the spaces between sets of 4 adjacent balls other balls rest, equal in size to the original. These balls form in turn a second layer on top of the first. Successive layers of this sort form a pyramidal pile with a single ball resting on top. If the bottom layer contains 16 balls, what is the height of the pile.

Result

h =  39.641 cm

Solution:

$r = 6.35 \ \\ (2x)^2 = (6r)^2+(6r)^2 \ \\ x^2 = 18r^2 \ \\ x = \sqrt{ 18 } \cdot \ r = \sqrt{ 18 } \cdot \ 6.35 \doteq 26.9408 \ \\ h_{ 1 }^2 +x^2 = (6r)^2 \ \\ h_{ 1 } = \sqrt{ (6 \cdot \ r)^2 -x^2 } = \sqrt{ (6 \cdot \ 6.35)^2 -26.9408^2 } \doteq 26.9408 \ \\ h = 2 \cdot \ r+h_{ 1 } = 2 \cdot \ 6.35+26.9408 \doteq 39.6408 = 39.641 \ \text{ cm }$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Pythagorean theorem is the base for the right triangle calculator.

Next similar math problems:

1. Inscribed circle
A circle is inscribed at the bottom wall of the cube with an edge (a = 1). What is the radius of the spherical surface that contains this circle and one of the vertex of the top cube base?
2. Tangent spheres
A sphere with a radius of 1 m is placed in the corner of the room. What is the largest sphere size that fits into the corner behind it? Additional info: Two spheres are placed in a corner of a room. The spheres are each tangent to the walls and floor an
3. Cube in sphere
The sphere is inscribed cube with edge 8 cm. Find the radius of the sphere.
Given that P = (5, 8) and Q = (6, 9), find the component form and magnitude of vector PQ.
5. Circle annulus
There are 2 concentric circles in the figure. Chord of larger circle 10 cm long is tangent to the smaller circle. What are does annulus have?
6. GP members
The geometric sequence has 10 members. The last two members are 2 and -1. Which member is -1/16?
7. Parametric equation
Find the parametric equation of a line with y-intercept (0,-4) and a slope of -2.
8. Third member
Determine the third member of the AP if a4=93, d=7.5.
9. Tetrahedral pyramid
It is given a regular tetrahedral pyramid with base edge 6 cm and the height of the pyramid 10 cm. Calculate the length of its side edges.
10. Church roof
The roof of the church tower has the shape of a regular tetrahedral pyramid with base edge length 5.4 meters and a height 5 m. It was found that needs to be corrected 27% covering of the roof area. What amount of material will be required?
11. Cards
Suppose that are three cards in the hats. One is red on both sides, one of which is black on both sides, and a third one side red and the second black. We are pulled out of a hat randomly one card and we see that one side of it is red. What is the probab
12. Theorem prove
We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
13. Euclid2
In right triangle ABC with right angle at C is given side a=27 and height v=12. Calculate the perimeter of the triangle.
14. Vector 7
Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
15. ABS CN
Calculate the absolute value of complex number -15-29i.
16. 4s pyramid
Regular tetrahedral pyramid has a base edge a=17 and collaterally edge length b=32. What is its height?
17. The ditch
Ditch with cross section of an isosceles trapezoid with bases 2m 6m are deep 1.5m. How long is the slope of the ditch?