Optimization Related Problem need ASAP

Question

A woman at a point A on the shore of a circular lake with radius r=3 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 4mph and row a boat at 2mph.

What is the shortest amount of time it would take her to reach point C?(hours)

What is the longest amount of time it would take her to reach point C following a path like the path in the figure?(hours)

Answer 1

Let us assume horizontal axis is between A and C. She starts at (-3,0) centre of the lake is at (0,0) and C is at (3.0).

Well suppose she walks clockwise for t hours, then in that time she walks 4t miles. As the circumference is 6π, she has walked through an angle of (4t/6π) x 360 degrees, lets call this a. Her x coordinate at that point will be 0 - cos (a) and her y coordinate will be sin (a)

Distance from point C then to row = √ ( 3+cos(a))² + sin² (a)

So total time T = t + 1/2 √ (3+cos a)² + sin² a

(a is a function of t)

So to minimise time, find t when dT/dt = 0