Who wins the Race?

Task

Alice and Briana each participate in a 5-kilometer race. Alice's distance covered, in kilometers, after $t$ minutes can be modeled by the equation $a(t) = \frac{t}{4}$. Briana's progress is modeled by the equation $b(t) = \sqrt{2t -1}$.  

1. Who starts first? Explain.
2. Who gets to the finish line first? Explain.
3. At what time(s) during the race are Alice and Briana side by side? Explain.

IM Commentary

The context in this task is a thin context, that is, a context that is not necessarily one that would arise in practice but that serves a mathematical purpose, namely to remind students that solving equation is about finding when two expressions have the same numerical value. 

The goal of this task is to solve an equation involving a radical and then verify whether the solutions of the resulting quadratic equation are relevant. In this particular case, one solution is relevant and the other is not because it occurs after the race is over. For this context, students can imagine that if the same equations continued to model Alice and Briana's race for an additional 5 kilometers, then they would meet again just over a half hour into the longer race.  

The model of Briana's progress might merit some discussion -- in addition to interpreting what it means that $b(t)$ is undefined for $t<\frac{1}{2}$ (the problem interprets this as saying that she has not started the race yet), it's also true that the model provides her an unrealistic speed at her entry into the race.  The teacher may wish to discuss this if the issue comes up. One context which might be appropriate for the given graphs is that Briana is on a skateboard while Alice is on foot. The beginning of the race goes downhill but then it levels off and perhaps goes uphill. This would partially explain Briana's average velocity for the first few minutes. The function is still too steep for values of $t$ between 0 and 2 but it might provide an overall good model for Briana's progress in the race.  Alternatively, the discussion might center around the process of modeling in general, and the inaccuracies it can produce.

Solution

1. The race starts when $t = 0$ and Alice starts off immediately as her distance traveled is positive as soon as $t \gt 0$. Briana does not start immediately as the expression $\sqrt{2t-1}$ is not defined when $2t -1 \lt 0$ or, in other words, when $t \lt \frac{1}{2}$. So Briana starts moving $\frac{1}{2}$ minute after Alice. 
2. The race is 5 kilometers so to see when Alice has finished the 5 kilometer race we can check when she has traveled 5 kilometers, that is we can find the value of $t$ with $a(t) = 5$. If $\frac{t}{4} = 5$ then $t = 20$ so Alice finished the race in 20 minutes. For Briana, we have $\sqrt{2t-1} = 5$ which means $2t -1 = 25$ or $t = 13$. It takes Briana only 13 minutes, after Alice has started, to complete the 5 kilometers: it takes Briana 12 $\frac{1}{2}$ minutes to complete the race measuring from the time she begins.  

3. To find out when Alice and Briana are together in the race, we can use the equations for how far they have traveled to identify for what times $t$ we have $a(t) = b(t)$. Squaring the given formulas for $a(t)$ and $b(t)$ gives $$\frac{t^2}{16} = 2t-1.$$ Manipulating this equation gives $t^2 -32t + 16 = 0$. We can solve this with the quadratic formula obtaining $$t = 16 \pm \sqrt{240}.$$ Alice and Briana are together after $16 - \sqrt{240}$ minutes (just a little over half a minute) and also after $16 + \sqrt{240}$ or about $31\frac{1}{2}$ minutes. Only the first value is relevant, however, as both Alice and Briana have finished by the 20 minute mark. This is depicted in the graph below (the independent variable here is $x$ rather than $t$):

   As shown in the graph and in the calculations above, Briana catches and passes Alice very quickly and, although she slows down as the race goes one, Alice only begins to catch up toward the end of the race and does not pass Briana. If they both continued and their progress were still modeled by these functions, then Briana would catch and pass Alice a little after one half hour.