A cube has 8 vertices, 12 edges, and 6 square faces. A soccer ball (also known as a buckyball or truncated icosahedron) has 12 pentagonal faces and 20 hexagonal faces. How many vertices and how many edges does a soccer ball have?
A circular table is placed in a corner of a room so that it touches both walls. A mark is made on the edge of the table, exactly 18 inches from one wall and 25 inches from the other. What is the radius of the table?
I'd really appreciate if someone could explain them to me instead of just giving an answer. I'd like to learn how to do them.
F+V-E = 2
If you count all the vertices of each polygon, each vertex is counted twice. So,
12+20 + (5*12+6*20)/2 - E = 2
E = 120
draw a diagram. If you draw horizontal and vertical lines through the point and through the center of the circle, you will see that
(r-25)^2 + (r-18)^2 = r^2
so, r = 13 or 73
I guess only 73 fits the real world, eh?
Thank you Steve. That was very helpful. I was the only one in my class that got those right thanks to you! (And at my school, doing that actually gets you bragging rights.)
Sure! Let's go through each question step by step.
Question 1:
To find the number of vertices and edges in a soccer ball (truncated icosahedron), we need to understand its structure. A truncated icosahedron consists of 12 pentagonal faces and 20 hexagonal faces.
To find the number of vertices, we can use Euler's formula for polyhedra, which states that the number of vertices plus the number of faces minus the number of edges equals 2.
In this case, the number of faces on a soccer ball is 12 pentagons and 20 hexagons, which adds up to 32 faces. The number of edges in a soccer ball is not given, but we can calculate it using the formula for the number of edges in a polyhedron.
The formula for the number of edges is: E = (V + F) - 2, where V represents the number of vertices and F represents the number of faces.
We know that a soccer ball has 32 faces, so we can substitute F = 32 into the formula. We also know that a soccer ball has 8 vertices, so we can substitute V = 8 into the formula.
E = (8 + 32) - 2
E = 40 - 2
E = 38
Therefore, a soccer ball has 8 vertices and 38 edges.
Question 2:
To find the radius of the circular table, we can use the concept of similar triangles. Let's form two triangles: one formed by the table and the wall it touches, and another formed by the mark on the table, the wall it touches, and the center of the table.
In the first triangle, we have a known side length of 18 inches, representing the distance from the table to one wall. In the second triangle, we have a known side length of 25 inches, representing the distance from the table to the other wall. Both triangles have a shared angle of 90 degrees.
By comparing the two triangles, we can set up a proportion to find the length of the unknown side in the second triangle, which represents the radius of the table. The proportion is:
(18 inches) / (x inches) = (25 inches) / (x + 25 inches)
By cross-multiplying, we get:
18(x + 25) = 25x
Simplifying the equation:
18x + 450 = 25x
Subtracting 18x from both sides:
450 = 7x
Dividing both sides by 7:
x = 64.29
Therefore, the radius of the table is approximately 64.29 inches.