Problem of the Week: Marathon Monday Edition

Happy Marathon Monday to all of our followers! It’s been a busy weekend for the Center of Math. We had a very successful couple of days at the NCTM 2015 conference! It was three days packed full of math people, thanks to each and every one of you for stopping by to chat with us. And since Friday afternoon, Boston has been buzzing with tourists, runners, and locals alike as everyone gets pumped for the marathon. In fact, I might have to alter my route home after work depending on when the runners finish.

For today’s Problem of the Week, which I posted on our Facebook, Twitter, and Google+ as usual,   I wanted to do a marathon theme. I created the problem below, which I adapted from this Washington Post blog.  

And as always, Monday comes with my messy handwriting providing the solution below...

Click the pictures to expand!

This first section was very straightforward. We simply change the hours:minutes format to just minutes, and divide our result by the distance of a marathon. 12.14 minutes per mile is quite respectable to those of us who aren't marathoners- it's just barely slower than five miles per hour. 

 For part B, we simply divided our total time in minutes by the time of a single interval to get the number. We're assuming that Zach is always running the first two minutes of the interval, and walking the third, so our result is a little silly- it has Zach finishing the marathon at a walk! If this was real instead of just a scenario, we know Zach could have pushed a little harder to complete the marathon at a run :)

Once we got the total number of intervals, we knew how many minutes Zach spent walking. And we multiplied the number of intervals by two to get the total minutes Zach spent running.

Part C was a little more advanced and a little more fun. We did some quick math to find how many minutes had passed when Meb crossed the finish line. The next step was to find what interval Zach had just completed when Meb finished the race. We find that Zach has completed 42 and 2/3s intervals, so he's completed the running section of his 43rd interval.

Then we need to identify where Zach was when he finished the 42nd interval. I did this by taking the total marathon distance (26.2 miles) and multiplying it by the fraction of 42/106 intervals, and got 10.38 miles for the first part of Zach's answer.

The rest isn't too difficult because we know that the 2/3 interval is all spent running, and we know that Zach runs twice as fast as he walks. So I drew the little diagram to demonstrate that 2/3 of the interval's time covers 4/5 of the interval's distance, or 4/5 of the interval distance of 0.2472 miles. So, when Meb crossed the finish line in only 2 hours 8 minutes, Zach was at 10.58 miles with a long way to go.

If you have any questions about my work, please leave a comment! I hope everyone gets to enjoy the marathon on TV or in person, and have a great start to the week.

-Tori