A right pentagonal prism has 15 edges in total. Using basic properties of prisms, determine how many vertices the prism has.

Introduction / Context:
This question involves basic solid geometry and properties of prisms. It asks you to relate the number of edges of a pentagonal prism to its number of vertices. Understanding how edges, faces, and vertices relate for prisms is useful in many combinatorial and geometry problems.

Given Data / Assumptions:

• The solid is a pentagonal prism, so the base is a pentagon.
• Total number of edges = 15.
• We must find the number of vertices.
• The prism is assumed to be a standard prism with two congruent parallel polygonal faces and rectangular lateral faces.

Concept / Approach:
For an n sided prism (where the base is an n sided polygon):

• Number of vertices = 2 * n (n on the top face and n on the bottom face).
• Number of edges = 3 * n (n edges on the top base, n edges on the bottom base, and n vertical lateral edges).

Here, we are told that the prism has 15 edges, so 3 * n = 15. From this we can find n, and then use 2 * n to get the number of vertices.

Step-by-Step Solution:
Step 1: Use the relation for edges of a prism: number of edges = 3 * n. Given that number of edges = 15, we have 3 * n = 15. Step 2: Solve for n: n = 15 / 3 = 5. Step 3: This means the base is a pentagon (5 sided polygon), which is consistent with a pentagonal prism. Step 4: Number of vertices for an n sided prism = 2 * n. So number of vertices = 2 * 5 = 10.

Verification / Alternative check:
We can visualise the prism. A pentagon has 5 vertices. A pentagonal prism has one pentagon on the top and one on the bottom, so there are 5 + 5 = 10 vertices. The edges consist of 5 edges on the top pentagon, 5 on the bottom pentagon, and 5 vertical edges joining corresponding vertices. That is 5 + 5 + 5 = 15 edges, which matches the given information. This confirms that our reasoning is consistent and the number of vertices is indeed 10.

Why Other Options Are Wrong:
Option A (12): This would be the number of edges for some other solids but not the number of vertices for this prism. Option C (15): This repeats the number of edges, not the vertices, and confuses two different counts. Option D (20): This number is too large and does not fit the formula for vertices in an n sided prism. Option E (8): This is fewer than the vertices of even a simple cube, so it is not logical for a pentagonal prism.

Common Pitfalls:
Students sometimes confuse the formulas for faces, edges, and vertices, or they try to apply Euler's formula without first identifying n. Another mistake is to treat the pentagonal prism as if it were a pyramid, which has a different pattern of edges and vertices. Remember that for a prism with n sided bases, the basic relationships are very regular: vertices = 2 * n and edges = 3 * n.

Final Answer:
The pentagonal prism has 10 vertices.