Task
Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.

Mike guesses that the current is flowing at a speed of 50 feet per minute. Assuming this is correct, how long will it take for Mike to reach his destination?

Mike does not really know the speed of the current. Make a table showing the time it will take him to reach his destination for different speeds:
Speed of Current
(feet per minute) 
Mike’s Speed
(feet per minute) 
Time for Mike to travel 30,000 feet (minutes) 
0 


50 


100 


140 


149 


$s$ 



The time $T$ taken by the trip, in minutes, as a function of the speed of the current is $s$ feet/minute. Write an equation expressing $T$ in terms of $s$. Explain why $s = 150$ does not make sense for this function, both in terms of the canoe trip and in terms of the equation.

Sketch a graph of the equation in part (c). Explain why it makes sense that the graph has a vertical asymptote at $s = 150$.
IM Commentary
The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols. Variation 1 of this task uses function notation and expects students to derive the formula for the function directly, without the aid of a table.
The task also provides an opportunity to discuss mathematical models, their interpretation, and their limits. For example, teachers could ask if it makes sense for $s$ to be negative. This might correspond to a flow of water moving in the same direction as Mike, and indeed the equation in the solution gives the correct answer in that case.
More fanciful, and requiring a longer discussion, is the question of whether it makes sense to consider values of $s$ larger than 150. If $s=300$, for example, a naive application of the formula predicts that Mike will arrive at his destination in $200$ minutes! It is reasonable to say that negative times do not make sense and to exclude values of $s$ greater than $150$. However, value $200$ could also be interpreted as referring to an event that takes place 200 minutes before the trip starts. If Mike had been at his destination 200 minutes ago, then a river which was flowing at 300 feet per minute against his direction of travel would push him precisely the 30,000 feet from his destination that the problem began with.
Comments
Log in to commentElizabeth says:
almost 3 yearsI like the task––nice use of context to support understanding of an asymptote. There is a typo in part c: The time T taken by the trip, in minutes, is a function of the speed of the current is s feet/minute. The second "is" should be changed to "of", or maybe the sentence should be revised to something like "The time T taken by the trip, in minutes, is a function of the speed s of the current, in feet/minute." More generally, I would love to have Word/editable versions of these tasks so that I could make minor changes to make them more classroom friendly. For example, I would make the header much smaller and reduce the font size of the header to fit the task on one page. Also, I would add space for student work and change the name of the protagonist. Finally, the variable "s" is problematic because it is easily confused with a "5" in handwritten work. I would probably use "c" for current. Thanks for providing these resources!
Ashli says:
almost 3 yearsHi Elizabeth, thank you for catching the typo! I believe the 'is' was meant to be an 'as'. I've gone ahead and fixed it and I think it reads better now. Also agreed that having the ability to do some edits or different print options would be great and it's on our wish list for the future. We appreciate knowing that there are folks in the IM community calling for the feature!  Ashli