Page Bird

The Tortoise and the Hare

A tortoise and a hare start at the same time in a 10-mile race. The hare runs for 20 minutes at 12 mph; stops to rest; falls asleep for several minutes; awakes; and finishes the race, again running at 12 mph. Meanwhile, the tortoise plugs along at the same constant rate of 2 mph throughout the race. What is the least amount of time the hare was asleep if the tortoise wins the race?

I started by plotting the tortoise's rate since he travels a constant rate.

The red line is the finish line at 10 miles. We can see that the tortoise will finish in five hours.

The hare on the other hand runs at 12 mph for only 20 minutes. Using the dilation and translation function on GSP I created a segment that represented this rate.

At this point, time is ticking away, but the hare is napping so all that needed to be done was to play around with the minimum amount of time he could nap if he is to lose. Using GSP, I changed the origin to be a floating y-value and x to be stationary at four miles.

Looking at the graph, we can see that if the hare wakes at 4.51 or about 4 hours and 30 minutes, then it will finish at exactly the same time as the tortoise. If the hare finishes at the same time as the tortoise, then the hare would have napped 4.51-0.33 = 4.18 hours or 4 hours 10 minutes and about 48 seconds. So if the hare sleeps longer than this then he will lose the race.
Another way to say this is the minimum amount of time the hare can sleep and lose the race is 4 hours 10 minutes and 49 seconds.