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?
I was given a similar problem a long time ago. I don't recall the solution (or method), but do remember an interesting experiment that was described along with it. The details are probably a little off, but this is the idea:
Apparently, some scientific-minded person took his dog (that was an experienced swimmer) out on a boat some distance from shore. He placed his son randomly along the shoreline, and had him call the dog. The dog eagerly swam to shore, and then ran along the shore to the boy. Amazingly, time after time, the angle taken by the dog (and thus its landing point) was EXTREMELY close to the optimum at which you arrive by analyzing the problem, knowing the dog's running and swimming speeds. (The dog did not swim straight to shore, nor did it swim straight to the boy). If nothing else, this would be fun to try.
Just a comment
