Differentiation

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  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

at 2 mph, it would take 2 hours to go the 4 mile distance directly across the lake is the shortest distance and shortest time.

at 4 mph, it would take 4pi/4 = pi hours to go around the lake. the semi-circumference is pi(r) = 4pi= about 12.57 miles

about 3.14 hours walking, 2 hours rowing

You could walk half way around the lake (or 1/4 way) then row the rest of the way. Or any combination of rowing and walking, but the time would be 2<t<3.14 hours