I’ve been going to the gym lately which is torture and I hate it. I am sore everywhere. I don’t understand why anyone finds it pleasurable to go there, but I do understand not wanting the body of a 70 year old when I’m 35, so I force myself. Anyway, while I was wheezing and sputtering on the treadmill yesterday I was inspired to write a challenge question, which I humbly present to you now. Winner gets a Math Guide.

 Note: Figure not drawn to scale.

The flat running surface on a certain treadmill is 5 feet long (delineated by the dashed lines in the figure). Two wheels, each with radius 4 inches, keep the belt moving smoothly as the user runs. Someone painted a yellow stripe across the belt because reasons. Mike counts the number of times he sees the yellow stripe while he runs to try to distract himself from the agony inherent in the activity. If he crosses the yellow stripe 1111 times in 20 minutes, what is his average speed over that time, in miles per hour? Round your answer to the nearest tenth.

UPDATE: Jayce got it first, and a book is merrily making its way to his abode. The solution is now posted below the cut.

The first thing we need to do is figure out how long the belt is. It’s a loop, basically, that has 2 straight sides, and two semicircular sides.

The straight sides are 5 feet long. That’s easy. The radius of each semicircular part is 4 inches, which means we need to convert to feet or we’re going to be in big trouble.

4 inches = 1/3 foot. The semicircular arc on each side, then, will be half the circumference of a circle with radius 1/3 foot. C = 2πr = (2/3)π. Since each side is only half the circle, each side is (1/3)π. Like so:

So the length of the treadmill belt is 10 + (2/3)π feet.

If we want to know how far I ran in 20 minutes, we now have the information necessary to figure it out.

If I saw the stripe 1111 times, that means I travelled 1111(10 + (2/3)π) feet. That’s roughly 13436.9 feet, but of course we should keep our numbers precise until the final step.

To convert to miles, divide the above result by the number of feet in a mile, 5280:

So that’s how far I ran in 20 minutes. At this point, I should come clean and admit that when I go to the gym, I don’t usually make it that far in 20 minutes. But I’m getting closer.

To get a speed in miles per hour, we need to multiply that by 3, since 20 minutes is only 1/3 of an hour:

There you have it. Rounded to the nearest tenth, that’s a speed of 7.6 miles per hour.