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?

Answer 1

t the woman row part way at an angle Θ to the diameter,
and walk along the arc for the rest of the way.

since the triangle on the diameter is right angled, angle subtended at the centre by the arc walked will be 2Θ

distance rowed is 2rcosΘ = 4cosΘ
and length of the arc jogged = r•2Θ = 4Θ, Θ in radians,
so T(Θ) = 4cosΘ/5 + 4Θ/10
= 0.8cosΘ + 0.4Θ

T'(Θ) = -0.8sinΘ + 0.4
but T"(Θ) = -0.8cosΘ, so setting T'(Θ) to 0 will give a MAXIMA
so minimum time has to be at one of the 2 extrema (Θ=0 or pi/2)
T(0) = 0.8
T(pi/2) = 0.4pi/2, < T(0)
so least time = 0.4pi/2 hrs