Question

You are asked to help set up a historical display in the park by stacking some cannonballs next to a Revolutionary War cannon. You are told to stack them by starting with a triangle in which each side is composed of four touching cannonballs. You are to continue stacking them until you have a single ball on the top centered over the middle of the triangular base. a. How many cannonballs do you need? b. What type of closest packing is displayed by the cannonballs? c. The four corners of the pyramid of cannonballs form the corners of what type of regular geometric solid?

Short answer

a. You will need 20 cannonballs to form the pyramid. b. The type of closest packing displayed by the cannonballs is face-centered cubic (fcc) packing. c. The four corners of the pyramid of cannonballs form the corners of a regular tetrahedron.

Step-by-step solution

a. Calculate the number of cannonballs needed

To find out the number of cannonballs needed, let's analyze the structure of the pyramid. Each layer of the pyramid will have the shape of an equilateral triangle. Starting from the base and moving upwards: - Layer 1 (base layer) - An equilateral triangle with 4 cannonballs on each side. - Layer 2 - An equilateral triangle with 3 cannonballs on each side. - Layer 3 - An equilateral triangle with 2 cannonballs on each side. - Layer 4 (top layer) - A single cannonball. To find the number of cannonballs in each layer, we can calculate the number of cannonballs in an equilateral triangular shape: Layer 1: \(1 + 2 + 3 + 4 = 10\) cannonballs Layer 2: \(1 + 2 + 3 = 6\) cannonballs Layer 3: \(1 + 2 = 3\) cannonballs Layer 4: \(1\) cannonball Now, sum up the number of cannonballs in all layers: \(10 + 6 + 3 + 1 = 20\). So, you will need 20 cannonballs.

b. Type of closest packing displayed by the cannonballs

The closest packing displayed by the cannonballs is called face-centered cubic (fcc) packing. In this type of packing, the centers of the cannonballs form a 3D lattice with a regular tetrahedron and an octahedron in each unit cell. The fcc packing is one of the most efficient ways to pack spheres as it allows them to fill up approximately 74% of the available space.

c. Regular geometric solid formed by the four corners of the pyramid

The four corners of the pyramid of cannonballs form the corners of a regular tetrahedron. A tetrahedron is a polyhedron with four triangular faces, six edges, and four vertices. In a regular tetrahedron, all faces are equilateral triangles, and all edges and angles are congruent.