- #1
Daveyboy
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Homework Statement
Jane is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest the boat. She can row her boat at 5 mph and can walk at 3 mph. Where should she land her boat to reach the village in the least amount of time.
Homework Equations
I don't know how to make a diagram so I'll try to describe it as carefully as possible.
There are two legs of a right triangle one 2 miles and the other 6 miles. (distance to shore and shore to village).
Let x be the distance from where the two legs (from above) meet to where the boat lands.
Then the distance the boat travels is [tex]\sqrt{x^{2}+4}[/tex] by the Pythagorean theorem.
Then the distance walked is 6-x miles.
Now use time = distance/rate
also the times can be added together to find the total time it takes for the trip. I'll take the derivative and try to solve for 0 but the solution is not real.
I'm sure I have the strategy correct for this problem, and I'm very confident I have the derivative and algebra correct. I think I need to interpret my system differently though, because I feel like I am only off my a minus sign somewhere.
The Attempt at a Solution
T(x) = [tex]\frac{\sqrt{x^{2}+4}}{5}[/tex] +[tex]\frac{6-x}{3}[/tex]
T'(x) = [tex]\frac{x}{5(\sqrt{x^{2}+4}}[/tex] - [tex]\frac{1}{3}[/tex]
When I try to solve T'(x) = 0 I do not get real answers. This is a problem.