Question

you stand on the shore of a circular lake and you wish to reach the exact opposite your current position. you can swim 20 feet per minute and run 50 feet per minute. what path should you take to reach your destination as quickly as possible?

Answer 1

assuming you swim along a chord that subtends an angle θ, you

swim 2rsin(θ/2)
walk πr-rθ

total time is thus

t = 1/10 r sinθ/2 + 1/50 (πr-rθ)
= r/50 (π - θ + 5sin(θ/2))

for minimum time, we need dt/dθ=0
5/2 cos(θ/2) - 1 = 0
cos(θ/2) = 2/5
θ = 2.32

so, walk for (π-2.32)r feet and then jump in and swim directly to the destination.