Algebra help - a race around a regular polygon

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Discussion Overview

The discussion revolves around a problem involving two runners, Bert and Ernie, who are running around a regular polygon with an unknown number of sides (x), each side measuring 12 meters. The problem is to determine how far Ernie will have traveled when they meet, given that Bert runs at twice the speed of Ernie. The scope includes algebraic reasoning and mathematical modeling.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • Some participants suggest starting with the formula for speed as distance divided by time.
  • One participant proposes calculating the circumference of the polygon to understand the total distance around which they are running.
  • Another participant notes that if Ernie runs at 1 m/s, Bert would run at 2 m/s, and they explore how far they would have run after specific time intervals.
  • It is mentioned that since Bert runs twice as fast, he will have covered twice the distance of Ernie when they meet, leading to the conclusion that Bert will have run 2/3 of the perimeter while Ernie runs 1/3.
  • One participant calculates the total distance as the perimeter (P = 12x meters) and expresses the distances run by Bert and Ernie in terms of the perimeter.

Areas of Agreement / Disagreement

Participants generally agree on the relationships between the speeds and distances of Bert and Ernie, but there is no consensus on the specific calculations or methods to arrive at the final answer.

Contextual Notes

There are assumptions regarding the speeds of the runners and the time taken until they meet, which are not fully resolved. The dependence on the number of sides of the polygon (x) also introduces uncertainty in the calculations.

Who May Find This Useful

This discussion may be useful for students seeking help with algebraic problems involving rates, distances, and time, particularly in the context of geometric figures like polygons.

aileenmarymolon
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Bert and Ernie are running around a regular polygon with x sides, all of length 12m. They start from the same point and run in opposite directions. If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?
 
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Re: algebra help

aileenmarymolon said:
Bert and Ernie are running around a regular polygon with x sides, all of length 12m. They start from the same point and run in opposite directions. If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?

Hello and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
Re: algebra help

I don't really know where to begin with this question. I know that Speed=distance / Time and that is about it.
 
Re: algebra help

aileenmarymolon said:
I don't really know where to begin with this question. I know that Speed=distance / Time and that is about it.

Well... what is the circumference of the polygon? (Wondering)

Suppose Ernie runs with a speed of 1 m/s, then Bert runs with a speed of 2 m/s.
How far will they have run after, say, 10 seconds?
After x seconds?
And after 2x seconds?
 
First, if they start at the same time, they will have run for the same time when they meet. Since Bert runs twice as fast as Ernie, he will have run twice as far as Ernie. That means that Bert will have run 2/3 of the way around the track and Ernie 1/3.
 
aileenmarymolon said:
Bert and Ernie are running around a regular polygon with x sides, all of length 12m.
They start from the same point and run in opposite directions.
If Bert is twice as fast as Ernie, how far will Ernie have traveled when they meet?
Bert's speed is twice that of Ermie.
Hence, Bert's distance (for a particular time) is twice that of Ernie.

When they first meet, their total distance is the perimeter, P = 12x meters.

Bert's distance is \tfrac{2}{3}P.
Ernie's distance is \tfrac{1}{3}P

Therefore . . .
 

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