Optimization - Why is there a constant term in the time equation?

[Qube]

Optimization - Why is there a constant term in the time equation??

Mod note: The OP apparently answered his own question.

Homework Statement

1) A man can run at 8 km/hr and swim at 4 km/hr. He is currently 6 km from the shore of a lake, which is due east from him. He wants to get to a point 10 km south on the shore of his current position. How should he proceed?

Homework Equations

1) Pythagorean Theorem; speed is distance / time; time is distance / speed.

The Attempt at a Solution

http://i.minus.com/jb1uFVsmbSfNtE.jpg

My question is: intuitively, if the man ran 0 km on the shore and swam all the way, he would have swam the square root of 136 km - two legs of the triangle are 6 and 10.

sqrt136 km divided by a rate of 4 km/hr yields a time of 2.91 hours. Distance over speed = time.

However, T(0) yields a different time of 3.25 hours. How come? I see that when x = 0 in the equation T(x) the first term - the (6-x)/8 term doesn't go to zero. Did I set up the problem incorrectly?

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Wait, never mind. I think I see why now. If x = 0, that wouldn't be that he ran 0 km. That would actually imply the opposite just looking at the diagram I setup. It would imply he ran 6 km and then swam 10 km. And time would correctly be 6 km / 8 plus 10/4.

 

[lippyka]

So the constant term in the time equation implies that the man is running 6 km before he starts swimming.