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Suppose that 20 pillars of the same heig...

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is

A

180

B

210

C

170

D

190

Text Solution

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The correct Answer is:
To solve the problem of finding the total number of beams connecting the tops of non-adjacent pillars in a circular stadium, we can use the concept of diagonals in a polygon. ### Step-by-Step Solution: 1. **Understanding the Structure**: - We have 20 pillars arranged in a circular manner. Each pillar can be thought of as a vertex of a polygon with 20 sides. 2. **Identifying Non-Adjacent Pillars**: - Each pillar can connect to every other pillar except for itself and its two adjacent pillars. Therefore, for each pillar, the number of non-adjacent pillars is: \[ 20 - 1 - 2 = 17 \] - This means each pillar connects to 17 other pillars. 3. **Calculating Total Connections**: - Since each of the 20 pillars connects to 17 non-adjacent pillars, the total number of connections (or beams) can be calculated as: \[ 20 \times 17 = 340 \] - However, this counts each beam twice (once from each pillar), so we need to divide by 2 to get the actual number of unique beams: \[ \text{Total beams} = \frac{340}{2} = 170 \] 4. **Conclusion**: - Thus, the total number of beams connecting the tops of the non-adjacent pillars is **170**.
To solve the problem of finding the total number of beams connecting the tops of non-adjacent pillars in a circular stadium, we can use the concept of diagonals in a polygon. ### Step-by-Step Solution: 1. **Understanding the Structure**: - We have 20 pillars arranged in a circular manner. Each pillar can be thought of as a vertex of a polygon with 20 sides. 2. **Identifying Non-Adjacent Pillars**: ...
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