Question

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..

Answer 1

...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

@satellite73 @ganeshie8 @zepdrix

Answer 3

@dumbcow

Answer 4

@iambatman @AccessDenied

Answer 5

@jim_thompson5910 @dan815

Answer 6

@Loser66 @Whitemonsterbunny17 @timo86m

Answer 7

@adison0112 , good explanation is on other post so i wont repeat, however is a summary of solution

distance can be written in terms of angle theta
\[d = 10\cos \theta + 10 \theta\]
we know distance = rate * time
--> time = distance/rate

now we have a function for time
\[T = \frac{10}{3} \cos \theta + \frac{10}{5.5} \theta\]
take derivative and set equal to 0 to maximize function
\[\rightarrow -\frac{10}{3} \sin \theta + \frac{10}{5.5} = 0\]
solve for angle
\[\sin \theta = \frac{3}{5.5}\]
\[\theta = 33.06^o\]

Answer 8

@adison0112