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.

Correct answer:

h =  39.6408 cm

Step-by-step explanation:

$$\begin{gathered} r=6.35 \\ (2x{)}^2=(6r{)}^2+(6r{)}^2 \\ {x}^2=18r^2 \\ x=\sqrt{18}\cdot r=\sqrt{18}\cdot 6.35\doteq 26.9408 \\ {h}_1^2+x^2=(6r{)}^2 \\ {h}_1=\sqrt{(6\cdot r{)}^2-x^2}=\sqrt{(6\cdot 6.35{)}^2-26.9408^2}\doteq 26.9408 \\ h=2\cdot r+h_1=2\cdot 6.35+26.9408=39.6408\text{ cm} \end{gathered}$$

Try another example

Related math problems and questions:

• Distance of points
  A regular quadrilateral pyramid ABCDV is given, in which edge AB = a = 4 cm and height v = 8 cm. Let S be the center of the CV. Find the distance of points A and S.
• 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 and
• Two balls
  Two balls, one 8cm in radius and the other 6cm in radius, are placed in a cylindrical plastic container 10cm in radius. Find the volume of water necessary to cover them.
• Hexagon cut pyramid
  Calculate the volume of a regular 6-sided cut pyramid if the bottom edge is 30 cm, the top edge is 12 cm, and the side edge length is 41 cm.
• 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.
• Sphere and cone
  Within the sphere of radius G = 33 cm inscribe the cone with the largest volume. What is that volume, and what are the dimensions of the cone?
• Pyramid cut
  We cut the regular square pyramid with a parallel plane to the two parts (see figure). The volume of the smaller pyramid is 20% of the volume of the original one. The bottom of the base of the smaller pyramid has a content of 10 cm². Find the area of the
• Quadrangular pyramid
  Calculate the surface area and volume of a regular quadrangular pyramid: sides of bases (bottom, top): a1 = 18 cm, a2 = 6cm angle α = 60 ° (Angle α is the angle between the sidewall and the base plane.) S =? , V =?
• 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?
• Cube in sphere
  The sphere is an inscribed cube with an edge of 8 cm. Find the sphere's radius.
• Spherical cap
  The spherical cap has a base radius of 8 cm and a height of 5 cm. Calculate the radius of a sphere of which this spherical cap is cut.
• Cube and sphere
  Cube with the surface area 150 cm² is described sphere. What is sphere surface?
• Nice prism
  Calculate the cuboid's surface if the sum of its edges is a + b + c = 19 cm and the body diagonal size u = 13 cm.
• Regular quadrilateral pyramid
  What is the volume of a regular quadrilateral pyramid if its surface is 576 cm² and the base edge is 16 cm?
• Cube diagonals
  Determine the volume and surface area of the cube if you know the length of the body diagonal u = 216 cm.
• Height of pyramid
  The pyramid ABCDV has edge lengths: AB = 4, AV = 7. What is its height?
• Space diagonal angles
  Calculate the angle between the body diagonal and the side edge c of the block with dimensions: a = 28cm, b = 45cm and c = 73cm. Then, find the angle between the body diagonal and the plane of the base ABCD.