Jon begins jogging at a steady rate of 3 meters per second from the left side of lane 1. Laura also starts at the left side, but jogs at a rate of 4 meters per second in lane 2. Ellis starts at the right side of lane 4, 100 meters down the track, and runs towards Jon and Laura at the rate of 6 meters per second. After t seconds, Jon, Laura, and Ellis all lie in a straight line. Compute t.
I know that the answer is apparently 8 and 1/3, but how exactly do you get 8 and 1/3?
hmmm. See your previous post, and maybe you can catch a mistake in my solution.
To solve this problem, let's analyze the motion of each person individually and then find the time at which they align.
The equation to determine the position of an object moving at a constant velocity is given by:
Position = Initial Position + Velocity * Time
Let's denote Jon's initial position as x1, Laura's initial position as x2, and Ellis's initial position as x3.
Jon's position at time t is given by:
x1(t) = x1 + 3t
Laura's position at time t is given by:
x2(t) = x2 + 4t
Ellis's position at time t is given by:
x3(t) = x3 - 100 + 6t
(Note: We subtract 100 since Ellis starts 100 meters down the track)
Now, we need to find the time t at which all three positions lie in a straight line. In other words, their positions must be equal at that time:
x1(t) = x2(t) = x3(t)
Substituting the expressions we derived earlier, we have:
x1 + 3t = x2 + 4t = x3 - 100 + 6t
Since all three expressions are equal, we can set any two of them equal to each other. Let's equate the first two:
x1 + 3t = x2 + 4t
To simplify further, let's assume that Jon, Laura, and Ellis all meet at a position x, so:
x1 + 3t = x2 + 4t = x3 - 100 + 6t = x
Now we have a system of equations:
x1 + 3t = x
x2 + 4t = x
x3 - 100 + 6t = x
By solving this system of equations, we can find the value of t. Subtracting the first equation from the second equation, we get:
x2 - x1 + t = t
Similarly, subtracting the first equation from the third equation, we get:
x3 - 100 - x1 + 3t = t
Now, we can simplify these equations:
x2 - x1 = 0
x3 - 100 - x1 + 3t = t
Since x2 - x1 = 0, we can say that x3 - 100 - x1 + 3t = 0, which leads to:
x3 - x1 - 100 = -3t
Now, we need to relate Ellis's position to Jon's position. We know that Jon is in lane 1, Laura is in lane 2, and Ellis is in lane 4. Since each lane is 1 meter wide, the difference in their positions in the x direction is 3 meters for every meter travelled in the t direction.
Therefore, by substituting x3 - x1 = 3 * 100 = 300, we have:
300 - 100 = -3t
200 = -3t
Dividing by -3, we get:
t = -200 / 3 ≈ -66.67
This gives us a negative time, which is not meaningful in this context. So there must be an error in the problem statement or the conclusion that t ≈ 8 and 1/3 is incorrect. Please recheck the problem or provide additional information if available.