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!
Sure, I'd be happy to help you with your algebra homework problem.
a. To solve this problem, we need to express the number of meters each runner has run after t seconds. Let's start by analyzing their movements.
Jon is jogging down the middle of Lane #1 at a steady 3 meters/sec. This means that after t seconds, Jon will have run a distance of 3t meters.
Laura starts even with Jon in the center of Lane #2 but moves at 4 meters/sec. Hence, after t seconds, Laura will have run a distance of 4t meters.
Ellis is initially located 100 meters down the track in Lane #4 and is heading towards Jon and Laura at 6 meters/sec. This means that after t seconds, Ellis will have run a distance of 100 - 6t meters. (We subtract 6t because Ellis is moving towards Jon and Laura.)
b. Now let's consider the location of each runner as a point on a graph. In this case, we can use the x-coordinate to represent the time in seconds (t), and the y-coordinate to represent the distance in meters that each runner has covered.
So, for Jon, the coordinates of his point on the graph would be (t, 3t).
For Laura, the coordinates would be (t, 4t).
And for Ellis, the coordinates would be (t, 100 - 6t).
c. To determine if three points are on a line, we can check if the gradients between any two points are equal. In other words, if the ratio of the change in y-coordinates to the change in x-coordinates is the same for any two pairs of points, then the points are collinear.
Let's compare the gradients between the points of Jon and Laura, and Jon and Ellis:
Gradient between Jon and Laura = (4t - 3t) / (t - t) = t / t = 1
Gradient between Jon and Ellis = (100 - 6t - 3t) / (t - t) = (100 - 9t) / 0
Since we cannot divide by zero, the gradient between Jon and Ellis is undefined. This means that the points of Jon, Laura, and Ellis are not collinear.
Therefore, the answer to the question "After how many seconds will the runners lie in a straight line?" is that they will never lie in a straight line.
I hope this explanation helps you understand how to approach and solve similar algebra problems. Let me know if you have any further questions!