Help with what I think is a differential problem

[perjac]

Hi All,

Did my maths B exam the other day but was absolutely stumped by one question. Spent half an hour thinking about it only to give up.

Question:

A man on a kayak (K) is 3 kilometres out to sea from the nearest point, (O) of a straight beach. His destination (D) is 6 kilometres along the beach from O. The fastest he can paddle is 4 km/hr and his maximum walking speed is 5km/h. How far from O should he go ashore to reach his destination in the least possible time?

Anyways the question was alongside a whole heap of differential problems so I assume you would need to differentiate an equation, find the min SP and that should tell you the distance but I have no idea how to get to the equation or if I was even on the right track.

This was the only question I couldn't answer and it is driving me nuts.

Thanks,

Perjac

 

[HallsofIvy]

Let P be the point at which the kayak lands on the beach. Let x be the distance from O to P. By the Pythagorean theorem, he must have paddled a distance \sqrt{x^2+ 9} kilometers. At 4 km/hr, that will require (1/4)\sqrt{x^2+9} hours. He then has to walk 6- x kilometers. At 5 km/hr, that will require (1/5)(6- x) hours. The total time is (1/4)\sqrt{x^2+ 9}+ (1/5)(6- x) hours. You want to minimize that. Take the derivative with respect to x and set it equal to 0.