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# Help

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You run at a rate of 4miles per hour and your friend runs at a rate of 3.5miles per hour. Your friend starts running 10 minutes before you, and you run for a half hour on the same path. Will you catch up to your  friend. Use a graph to support your answer.

Sep 17, 2017

#1
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Here's a graph:

Sep 17, 2017
#2
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If your friend runs at a rate of 3.5 mph and has a 10 minute - 1/6 hr -  head start......he will be

3.5 * (10/ 60)  =  3.5 * (1/6)  = 7/12 of a mile ahead of you when you start

The distance he covers in one minute  = 3.5 / 60  =  7/ 120 miles

The distance the you cover in one minute  = 4/60   =  8/120 miles

So....the distance that you close on him 1/2 hour after you start running [ 30 minutes ]  =

7/12 - [8 - 7 ] / 120 * 30  =  7/12 - 1/4 = 1/3 mile   which is less than the 7/12 mile distance that you have to close

Here's a graph that shows the closing distance between you and your friend x minutes after you start running : https://www.desmos.com/calculator/pavi3luxdt

As the graph shows....you will catch you friend after 70 minutes when both of you have run 14/3 miles  [ 4 + 2/3 ] miles......at that point the closing distance you have to make up = 0 miles

Sep 17, 2017
edited by CPhill  Sep 17, 2017
#3
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In half an hour, your friend would have run:
3.5 x 1/2 hour=1.75 miles. And at the same time, you would have run:
4 x 1/3 hour =1 1/3 miles. So, you could not have caught him at the finish line.
Let the time when you caught up with him =t minutes. So you have:
3.5t = 4(t - 10), solve for t
t = 80 minutes - your friend's running time when both of you will meet. And: 80 - 10 = 70 minutes - your running time.

Sep 17, 2017