Question

A woman at a point A on the shore of a circular lake with radius r=4 wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 10mph and row a boat at 5mph. What is the shortest amount of time it would take her to reach point C?
I'm not sure where to start... :(

Answer 1

Attached is a drawing to set up the problem, hopefully that helps. 
The red lines are tracing the rowing and walking to get to other side.
dR = distance rowed
dw = distance walked
x = the angle in which to start rowing initially 

dR can be found by using Law of cosines
$dR = \sqrt{4^2 +4^2 - 2(4)(4) cos(180-2x)} = 4\sqrt{2}\sqrt{1-cos(180-2x)}$
dw is arc length, which is just radius*arc
$dW = 4*2x = 8x$

next we need t set up a function for Time in terms of x.
This is so we can minimize the time it takes to get to other side.
Time = distance/rate

$T = \frac{dR}{5} + \frac{dW}{10} = \frac{4 \sqrt{2}}{5} \sqrt{1-cos(180-2x)}+\frac{4}{5} x$
Finally, take derivative and set equal to 0
Once you solve for x, plug it back into Time function to obtain final answer.