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.
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
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.