Answer :
To solve the problem of determining the number of vertices in a regular pyramid with 40 edges, we can use Euler's formula, which is:
[tex]\[ V + F = E + 2 \][/tex]
where:
- [tex]\( V \)[/tex] is the number of vertices,
- [tex]\( F \)[/tex] is the number of faces,
- [tex]\( E \)[/tex] is the number of edges.
According to the problem, the number of vertices in the pyramid is equal to the number of faces, so we have:
[tex]\[ V = F \][/tex]
We are also given that the number of edges [tex]\( E \)[/tex] is 40.
Using these pieces of information, let's substitute into Euler's formula:
[tex]\[ V + F = E + 2 \][/tex]
[tex]\[ V + V = 40 + 2 \][/tex]
[tex]\[ 2V = 42 \][/tex]
Solve for [tex]\( V \)[/tex]:
[tex]\[ V = \frac{42}{2} \][/tex]
[tex]\[ V = 21 \][/tex]
Therefore, the number of vertices in the pyramid is 21.
[tex]\[ V + F = E + 2 \][/tex]
where:
- [tex]\( V \)[/tex] is the number of vertices,
- [tex]\( F \)[/tex] is the number of faces,
- [tex]\( E \)[/tex] is the number of edges.
According to the problem, the number of vertices in the pyramid is equal to the number of faces, so we have:
[tex]\[ V = F \][/tex]
We are also given that the number of edges [tex]\( E \)[/tex] is 40.
Using these pieces of information, let's substitute into Euler's formula:
[tex]\[ V + F = E + 2 \][/tex]
[tex]\[ V + V = 40 + 2 \][/tex]
[tex]\[ 2V = 42 \][/tex]
Solve for [tex]\( V \)[/tex]:
[tex]\[ V = \frac{42}{2} \][/tex]
[tex]\[ V = 21 \][/tex]
Therefore, the number of vertices in the pyramid is 21.
