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Word Problems : Relative Speed of Swimmers

(#48) Two swimmers start at opposite ends of a pool 89 feet long. One person swims at the rate of 19 feet per minute and the other swims at a rate of 53 feet per minute. How many times will they meet in 33 minutes?

(#33) Two swimmers start at opposite ends of a 90-foot pool. One swims 30 feet per minute and the other at 20 feet per minute. If they swim for 30 minutes, how many times will they meet each other?
My initial thought was to set the variables of the problem as follows:

I have answered two problems involving the distance of swimmers. My solutions are detailed in the attachment. My professor has emailed me the following statements in two separate emails:
EMAIL #1
RE: Q48 Could you draw a graph which shows each
swimmer.[distance(y axis) vs. time(x axis)]At point origin one swimmer at 0 distance the other one 89 since they are opposite side of the pool.

EMAIL #2
I need to see or hear your reaction when
you compare your (Swimmer problems)solutions with the graphing solutions.

I am not sure what I will find to be different after I graph them, but I am assuming he thinks something will be different or he wouldn't be asking me to do this. This is where I need your help. I need you to look at the two questions (attachment) with my responses and graph the solutions to each of these problems. If you find something different in these two problems, then I need you to explain what is different and why. I hope this makes sense. THANKS!

Attachments

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Please see the attached file for the complete solution.
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First of all, let me tell you about the mistake you had made in your solution. Let's consider the first question for example. You are right about the first time they meet. They have actually swum the length of the pool together at that time, but the next time they meet would be when the slower swimmer has not finished one length yet while the faster one overtakes and goes for the second length. They are in the same direction now. Therefore, whenever the swimmers meet in the opposite directions your assumption is correct, but when they meet in the same direction it will be violated.

1)
Ok, let's call the swimmer whose speed is 19: A; and the one whose speed is 53: B. Moreover, we consider the start point to be the point where A starts and the end point is the point where B starts. Obviously the dominant one here is B whose speed is higher. Well, we observe that:
tA=89/19≈4.684 min
tB=89/53≈1.679 min.
tA and tB are the times in which A and B swim from start to end (or end to start). We must assume that when A and B reach the start/end point each time, they quickly and without any change ...

Solution Summary

The number of times swimmers meet given speed and time are calculated.

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