Student Resources
Functions: Marathon Paradox
An interesting puzzle about rates:
Chort and Frey ran a marathon (26.2 miles). Chort ran at a perfectly uniform pace of eight-minutes-per-mile. Frey took exactly eight minutes and one second to complete each one-mile interval. This refers to all one-mile intervals, including, for example, the interval from 5.63 miles to 6.63 miles. Nevertheless, Frey finished ahead of Chort. Explain how.
This seemingly impossible result is in fact possible. Calculus is fundamentally about rate of change and this paradox illustrates that we need to understand those foundational concepts deeply if we are going to apply it throughout calculus!
The following graph presents one possible solution. Can you reconcile the details of the graph with those of the story?
Click ⛶ to open in full screen
Distance is measured in miles on the vertical axis.
Time is measured in minutes on the horizontal axis.
Describe Chort's motion as accurately and completely as possible. Describe Frey's motion as accurately as possible.
There is a horizontal line at the distance of 26.2 miles. Drag the graph and zoom in where they intersect this line. How much does Frey win by? What is the most time that Frey could win by using a modified version of his strategy?
Select the box by "Chort's Triangle." This shows corresponding changes in distance \(\Delta x\) and changes in time \(\Delta t\) for Chort's progress during the marathon. You can drag both red dots along his graph. What do you observe about the relationship between \(\Delta x\) and \(\Delta t\) on Chort's graph?
Select the box by "Frey's Triangle." This shows corresponding changes in distance \(\Delta x\) and changes in time \(\Delta t\) for Frey's progress during the marathon. You can only drag the lower blue dot along his graph. What do you observe about the relationship between \(\Delta x\) and \(\Delta t\) on Frey's graph? What would happen if you could drag the upper red dot on the triangle for Frey's graph?