Expert: Steve Holleran - 9/20/2007

Question
How do you set up a function in terms of time for the following: "You are in a canoe 3 miles from the nearest point on the shore of a lake.  You want to get back to your camp, which is 2 miles down the shore of the lake (from the nearest point) and 1 mile inland.  You can row at 2 mph and walk at 4mph." I understand how to draw the picture but I don't get how I can set up a system of equations to find a function using this information.

Answer
Hi Keith,

I don't know that you need a system of equations, but what I don't understand is whether this function is supposed to represent any path to the camp, or the one that takes the shortest time?

Because the path for the least amount of time would be a direct line from the canoe to the camp:

                               D (camp)
                               |
                               |
               A --------------|B
                 |
                 |
                 |
                 |
                 |C (canoe)
If you draw a line from C to D, and call the point where it crosses AB as E, then you have:

AE = x, so EB = 2-x

Then CE = sqrt(9 + (2-x)^2)

    ED = sqrt(1 + x^2)

Then T = D/R = Time on water + Time on land

       =  sqrt(9 + (2-x)^2)/3 + sqrt(1 + x^2)/4

This is the best I can come up with for this situation.
I hope it helps you out.

Steve Holleran