## 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.

Let $s$ be the speed of the current in feet per minute. Write an expression for $r(s)$, the speed at which Mike is moving relative to the river bank, in terms of $s$.

Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for $T(s)$, the time in minutes it will take, in terms of $s$.

What is the vertical intercept of $T$? What does this point represent in terms of Mike’s canoe trip?

At what value of $s$ does the graph have a vertical asymptote? Explain why this makes sense in the situation.

For what values of $s$ does $T(s)$ make sense in the context of the problem?

## IM Commentary

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task. Variation 2 of this task does not use function notation, and focuses more on the numerical and graphical behavior of the function near its vertical asymptote.

As a teaching task it 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.