Here's my question. Please walk me through it so I can fully understand it.
Jon begins jogging at a steady 3 meters/sec down the middle of Lane #1 of a public track. Laaura starts even with him in the center of Lane #2 but moves at 4 meters/sec. At the instant they begin, Ellis is located 100 meters down the track in Lane #4, and is headed toward them in his lane at 6 meters/sec. After how many seconds will the runners lie in a straight line?
a. Let t be the number of seconds the three have been running. Write expressions for the number of meters each has run after t seconds.
b. Consider the location of each runner as a point on a graph. What quantity might you use as the x-coordinate? What quantity might you use as the y-coordinate?
c. How can you tell if three points are on a line? Use this to solve the problem.
Any help is appreciated, even if you can only solve one part. Thank you!
Please read my answer of yesterday
Sure! Let's break down the problem step by step:
a. Let's start by writing expressions for the number of meters each runner has run after t seconds.
- Jon's distance can be calculated by multiplying his velocity (3 m/s) by the time (t) in seconds. So, Jon's distance is 3t meters.
- Laura's distance can be calculated in the same way. Since her velocity is 4 m/s, Laura's distance is 4t meters.
- Ellis is already 100 meters down the track, so his distance can be calculated by taking 100 meters and subtracting his velocity (6 m/s) multiplied by the time (t) in seconds. Ellis' distance is 100 - 6t meters.
b. We can consider the location of each runner as a point on a graph and use x-coordinates and y-coordinates to represent their positions.
- Let's take the x-coordinate to represent time (t) in seconds. This will allow us to track how the position of each runner changes over time.
- For the y-coordinate, we can use the distance each runner has covered. So, Jon's y-coordinate would be 3t, Laura's y-coordinate would be 4t, and Ellis' y-coordinate would be 100 - 6t since he is moving towards the middle.
c. To determine if three points are on a line, we can check if the slope between any two points is equal. If the slopes are equal, the three points lie on a straight line.
- Let's consider any two runners at a time. If Jon and Laura are on a straight line, then the slope between their points (y2 - y1) / (x2 - x1) should be equal to the slope between Laura and Ellis' points.
- So, (4t - 3t) / (t - 0) should be equal to (100 - 6t - 4t) / (t - 0) for the three points to be on a straight line.
- Simplifying the equation, we get (t) / (t) = (100 - 10t) / (t), which can be simplified to 1 = 100/t - 10.
- By multiplying both sides by t, we get t = 100.
Therefore, after 100 seconds, all three runners will lie in a straight line.
I hope this helps! Let me know if you have any more questions.