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How many vertices does a regular pyramid with 40 edges have?

Use Euler's formula: [tex]V + F = E + 2[/tex]

A. 20
B. 21
C. 40
D. 42

Answer :

To solve this problem, we can use Euler's formula for polyhedra, which states:

[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

For a pyramid, we are given that the number of vertices is equal to the number of faces, so [tex]\( V = F \)[/tex].

We are also given that the pyramid has 40 edges [tex]\((E = 40)\)[/tex].

Substitute the known values into Euler's formula:

[tex]\[ V + F = 40 + 2 \][/tex]

Since [tex]\( V = F \)[/tex], we can express everything in terms of [tex]\( V \)[/tex]:

[tex]\[ V + V = 40 + 2 \][/tex]

[tex]\[ 2V = 42 \][/tex]

To find the number of vertices ([tex]\( V \)[/tex]), divide both sides of the equation by 2:

[tex]\[ V = \frac{42}{2} = 21 \][/tex]

Therefore, the regular pyramid with 40 edges has 21 vertices.

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