Distance problem. - linear equation

In summary, if A and B start at the same place and run in the same direction, it will take the slower runner 3.75 minutes to pass the faster runner. If they start at different places and run in opposite directions, the slower runner will take 4.15 minutes to pass the faster runner.
  • #1
paulmdrdo1
385
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A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the same instant from the same place, in how many minutes will they pass each other if they run around the track (a) in the same direction, (6) in opposite directions?

I know how to solve b my answer is 3.75 mins.

but I don't get the first question. I can't clearly picture the scenario in my mind for the 1st question. can you provide an explanation for it. thanks!
 
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  • #2
For (a), think about what happens after 10 minutes. At that point, the slow runner will have completed one lap, but the faster runner will already be two-thirds of the way round the second lap, and before long will overtake the slower runner. It's that overtaking moment that you are looking for.
 
  • #3
how did you know that after 10 mins the faster runner will be two-thirds of the way round the second lap?
 
  • #4
paulmdrdo said:
how did you know that after 10 mins the faster runner will be two-thirds of the way round the second lap?
Because he/she will have completed the first lap in 6 minutes and will then have had another 4 minutes, in which time s/he will have completed 4/6 of a lap.
 
  • #5
paulmdrdo said:
A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the same instant from the same place, in how many minutes will they pass each other if they run around the track (a) in the same direction, (6) in opposite directions?

I know how to solve b my answer is 3.75 mins.

but I don't get the first question. I can't clearly picture the scenario in my mind for the 1st question. can you provide an explanation for it. thanks!
Here's how I would do it. Since the track is one mile long, A is running at 1/6 mile per minute and B is running at 1/10 mile per minute. A's speed, relative to B is 1/6- 1/10= 5/30- 3/30= 2/30= 1/15 mile per minute. To pass B, A must have run one mile more than B in the same time. At 1/15 mile per minute, it would take A 15 minutes to do that.

When they run in opposite directions, so they are "closing" on each other, their relative speeds are 1/6+ 1/10= 5/30+ 3/30= 8/30= 4/15 mile per minute. To go one mile at 4/15 mile per minute would require 15/4= 3.75 minutes as you say.
 
  • #6
what do you mean by "A's speed, relative to B is 1/6- 1/10= 5/30- 3/30= 2/30= 1/15 mile per minute."?
 
Last edited:
  • #7
why does A have to be 1 mile more than B to overtake it? can you explain?
 
  • #8
that's also my question. anybody? please answer.
 
  • #9
How long is one lap?
 
  • #10
the circular lap is one mile. It still can't picture it clearly. this is one of the problems that always get me scratching my head.
 
  • #11
Okay, since the lap is one mile long, then when the faster runner laps the slower, he/she has then run exactly one mile more. :D
 
  • #12
(Dance)(Bow)(Blush) oh men! It seems I should get some rest now. To boost my critical thinking. Now I understand it clearly! Thanks!
 

1. What is a distance problem in terms of linear equations?

A distance problem involves finding the distance between two points on a coordinate plane using a linear equation. This can be done by plugging in the x and y coordinates of the two points into the equation and solving for the distance using the Pythagorean theorem.

2. How do you set up a linear equation to solve a distance problem?

To solve a distance problem using a linear equation, you need to first identify the two points on the coordinate plane. Then, choose one point to be the origin and use the distance formula to set up an equation with the other point as the endpoint. This will typically involve using the Pythagorean theorem to find the length of one of the sides of a right triangle.

3. Can a distance problem have more than two points?

Yes, a distance problem can have more than two points. The distance formula can be used to find the distance between any two points on a coordinate plane, whether there are two or more points involved.

4. What are some real-life examples of distance problems?

Some real-life examples of distance problems include finding the distance between two cities on a map, calculating the distance traveled by a moving object, or determining the length of a diagonal fence in a rectangular yard. These types of problems can be solved using linear equations and the distance formula.

5. Are there any other methods for solving distance problems besides using linear equations?

Yes, there are other methods for solving distance problems, such as using geometry or trigonometry. These methods may be more appropriate for certain types of distance problems, such as those involving angles or curved lines. However, linear equations are often the most efficient and straightforward method for solving distance problems on a coordinate plane.

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