A man is sitting in a lake in his boat fishing when he receives a call on his cell phone. A barbecue is happening a ways down the shoreline, and he had better get there fast so as not to miss out. He is two miles out perpendicular to the shore, and 7 miles horizontally from the location on the beach from the barbecue. If the lake has no current and the wind is negligible, he can row toward the shoreline at a rate of 3 mph. When he reaches dry land, he can run at 5 miles per hour. If he wants to reach the barbecue as quickly as possible, how far horizontally should he land the boat from his current location?

As a bonus, if we assign the distance from the shore to be A miles, the distance from the barbecue along the shoreline B miles, and the boat speed and running speed C and D miles per hour respectively, does there exist a function that will output the ideal place to land the boat for all positive values of A,B,C, and D? If so, what is it? If not, why not?

(In reply to Solution by Bractals)

Very nice.  I'm a bit rusty on my Calculus, but I eventually ended up with the same answer.  Good problem, too.

As for the case where C>=D, the solution is a beeline to the barbeque site without leaving the lake.  Let's eat!

Posted by MindRod on 2005-12-05 21:53:13