How do I solve this problem?

Question

You paddle a canoe at a steady speed on a river where the current’s speed is 1km/hr. You go 7.7km downstream and then 7.7 km upstream to get back to where you started. The trip takes 3.6 hours.

a. Write a rational equation that models the problem. State what the variable in your equation represents.

b. Solve the equation and state the solution in terms of the context of the problem.

Answer 1

This is a tricky problem, but I think we can work through it. The main idea we are going to use here is that distance = speed x time. We have two trips to think about, the upstream trip and the downstream trip. Even though we paddle at the same speed against the water on both trips, the current in the river means the accual speed of the boat (relative to the ground) will be different for each trip. Thus, the time for each trip will be different as well. Let's call the speed we can paddle against the water 's'. Then, since the current is 1 km/hr, our downstream speed is 's + 1' and our upstream speed is 's - 1'. This is enough to start modeling the problem. Since there are two trips, it is going to take two equations, one for upstream and one for downstream. Here they are:

7.7 = (s + 1) * t_d

7.7 = (s - 1) * t_u

These are both distance = speed * time equations. We still have three unknown quantities here, so we need to use the last bit of given information. Since the total time for both trips is 3.6 hrs we know that t_u = 3.6 - t_d. We can substitute this into the second equation to get:

7.7 = (s + 1) * t_d

7.7 = (s - 1) * (3.6 - t_d)

Now we have two equations with two unknowns and we are on our way. Solving these equations will tell us how fast we can paddle and how long the downstream trip took. I'll be checking in all day, so let me know if you need some help solving this system of equations.