Question

Optimization problem,

Answer 1

A circular lake has a diameter of 10 miles. Mathman and several other contestants are assembled at point A and need to get to point b in the shortest amount of time. The rules are that you can get into a boat and row in a straight line towards any point P on the circumference between A and B. You then get out of the boat and run along the remaining circumference until you get to point B. Each contestant can row a boat at 3 miles per hour and can run around the circumference of the lake at 5.5 miles per hour.After a few quick calculations Mathman climbs in his boat, heads in a direction that makes angle theta with diameter AB, gets out at point P and runs around the circumference to point B. To Mathman's dismay, all the other contestants have arrived before him. Mathman then realizes that his calculations gave him the angle theta which made for the longest trip, not shortest. What angle theta did Mathman use?

Answer 2

@ganeshie8 @satellite73 @Ashleyisakitty

Answer 3

@Destinymasha @undeadknight26 @iambatman

Answer 4

@Firejay5

Answer 5

Is AB the diameter, or is B arbitrarily placed on the lake shore?

Answer 6

ab is the diameter, i think that b is also the finish line

Answer 7

Okay, according to Thales' theorem, the angle APB is always a right angle because one of the chords is a diameter: http://en.wikipedia.org/wiki/Thales'_theorem#First_proof

The total distance traveled is the length of the chord AP plus the length of the arc PB. If we denote \(x\) to mean the arc measure and \(L\) the arc length, then we can find the optimal \(\theta\). Presumably, \(0\le\theta\le\dfrac{\pi}{2}\). Any more or less would mean the angle is larger or smaller than can be allowed.