A solid has faces that consist of 4 triangles, 3 rectangles, and 1 hexagon. The solid has 9 vertices. How many edges does the solid have?
A.
12 edges
B.
15 edges
C.
18 edges
D.
21 edges
The number of edges can be found using Euler's formula for polyhedra:
Number of vertices + Number of faces - Number of edges = 2.
Given that the solid has 9 vertices, 4 triangles, 3 rectangles, and 1 hexagon, we can substitute these values into the equation:
9 + 4 + 3 + 1 - Number of edges = 2.
Simplifying this equation gives:
17 - Number of edges = 2.
Adding "Number of edges" to both sides of the equation gives:
17 = 2 + Number of edges.
Subtracting 2 from both sides of the equation gives:
Number of edges = 17 - 2 = 15.
Therefore, the solid has 15 edges.
Answer: B. 15 edges.
To determine the number of edges in the solid, we can use Euler's formula, which states that for any polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the equation V + F = E + 2.
Given that the solid has 9 vertices, let's determine the number of faces. Since each face consists of 4 triangles, 3 rectangles, and 1 hexagon, we have a total of 4 triangles + 3 rectangles + 1 hexagon = 8 faces.
Substituting the values into Euler's formula: 9 + 8 = E + 2.
Simplifying the equation: 17 = E + 2.
Now, subtracting 2 from both sides: 17 - 2 = E.
Therefore, the solid has E = 15 edges.
Hence, the correct answer is option B: 15 edges.