Question

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!

Answer 1

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

Answer 2

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