I'm having some trouble with my algebra homework. Please walk me through the question completely and give me the answer so I can fully understand it if I would ever need to solve a similar problem.
Here's the problem:
#1: Jon begins jogging at a steady 3 meters/sec down the middle of Lane #1 of a public track. Laura 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 heading towards 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.
Thanks so much in advance. Any help is appreciated, even if you can only solve one part. Thanks!
The location of J from the starting line is 3 t. The location of L is 4t. The location of E from the starting line is 100 - 6t.
A straight line from J through L intersects the middle of lane 4 at a distance 6t from the starting line. (Remember that lane #3 is empty).
When 6 t = 100 - 6t, the runners are on a straight line.
12 t = 100
t = 8 1/3 seconds
At that time XYjon (lane 1) = 25 m; Ylaura (lane 2)= 33.33 m and Yellis (in lane 4) = 50 m
Make a plot of showing where the runners are in X (lane number) and Y (from starting line) at that time and you will see they fall along a straight line
The location of Jon can be written as (3t, 1), with the x-coordinate being the position he is from the starting line and the y-coordinate being the lane number he's in. If you do this with Laura and Ellis, their positions will be (4t, 2) and (100-6t, 4) respectively. Since collinear points have the same slope, you can solve for the slope between the three points and set them equal to each other. For example:
m = (2 - 1) / (4t - 3t), which will give you the slope 1 / t.
m = (4 - 2) / (100 - 6t - 4t), which will give you the slope 2 / 100 - 10t.
If you set them equal to each other as a proportion and solve for t, you'll get t = 8.3
Sure! I'll walk you through each part of the problem and help you solve it step by step.
a. Let's start by writing expressions for the number of meters each runner has run after t seconds.
- Jon: Jon is jogging at a steady 3 meters/sec. So, after t seconds, he would have run 3t meters.
- Laura: Laura is moving at 4 meters/sec. So, after t seconds, she would have run 4t meters.
- Ellis: Ellis is located 100 meters down the track in Lane #4 and is heading towards Jon and Laura at 6 meters/sec. So, after t seconds, he would have run 6t meters in the opposite direction. Therefore, we need to subtract 6t from 100 to get his position in relation to Jon and Laura.
b. We can consider the location of each runner as a point on a graph. For the x-coordinate, we can use the time in seconds (t) since the runners started. For the y-coordinate, we can use the number of meters each runner has covered after t seconds.
- Jon's position in the graph would be (t, 3t)
- Laura's position would be (t, 4t)
- Ellis' position would be (t, 100 - 6t)
c. To determine if three points are on a line, we need to check if the slopes between any two pairs of points are equal. If all the slopes are equal, then the three points lie on the same line.
In this problem, both Jon and Laura are moving at a constant speed, so their positions (3t, 4t) will always lie on a straight line. The line connecting their positions will have a slope of (change in y)/(change in x) = (4t - 3t)/(t - t) = t/t = 1.
Similarly, Ellis' position (100 - 6t, t) will also lie on a straight line with a slope of (change in y)/(change in x) = (t - t)/(100 - 6t - 100) = 0.
Since the slopes of the lines connecting the runners are different (1 and 0), the three runners will never lie on a straight line.
Therefore, there is no solution to the problem, and the answer is that the runners will never lie in a straight line.
I hope this helps you understand how to approach and solve this problem! Let me know if you have any further questions.