# Canoe Trip

Alignments to Content Standards: A-REI.A.2

Jamie and Ralph take a canoe trip up a river for $1$ mile and then return. The current in the river is 1 mile per hour. The total trip time is 2 hours and 24 minutes. Assuming that they are paddling at a constant rate throughout the trip, find the speed that Jamie and Ralph are paddling.
Suppose we let $x$ denote the speed, in miles per hour, that the canoe would travel with no current. When they are traveling against the current, Jamie and Ralph's speed will be $x-1$ miles per hour and when they are traveling with the current their speed will be $x+1$ miles per hour. The trip upstream will take $\frac{1}{x-1}$ hours and the trip downstream will take $\frac{1}{x+1}$ hours. There are $\frac{2}{5}$ of an hour in 24 minutes so the total trip lasts for $2\frac{2}{5}$ hours giving us $$\frac{1}{x-1}+ \frac{1}{x+1} = \frac{12}{5}.$$ Multiplying both sides of the equation by $(x-1)(x+1) = x^2-1$ gives $$(x+1) + (x-1) = \frac{12}{5}(x^2-1).$$ This equation simplifies to $\frac{12}{5}x^2 -2x - \frac{12}{5} = 0$ or, after further manipulation, $$6x^2 - 5x - 6 = 0.$$ We can use the quadratic formula to solve for $x$: $$x = \frac{5 \pm\sqrt{25+144}}{12}.$$ We have $\sqrt{169} = 13$ so the two solutions are $x = \frac{5\pm13}{12}$ or $x = \frac{3}{2}$ and $x = -\frac{2}{3}$. The second solution does not make any sense in this context as the speed cannot be negative. So Jamie and Ralph are paddling at a rate of $\frac{3}{2}$ miles per hour. Going upstream, the trip takes longer against the current and going with the current the trip is shorter.