Question

The shoreline of a lake is a circle with diameter 3 km. Peter stands at point E and wants to reach the diametrically opposite point W. He intends to jog along the north shore to a point P and then swim the straight line distance to W. If he swims at a rate of 3 km/h and jogs at a rate of 24 km/h. How far should he jog in order to arrive at point W in the least amount of time?

Answer 1

I made a diagram showing a diagonal WE
I placed P along an arc called a so that the central angle is 2Ø
I joined PW and let PW = x
Recall that the central angle subtended by an arc is twice the angle subtended at the circle, so angle PWE = Ø

recall arc = r x central angle
a = 1.5(2Ø) = 3Ø

also in the internal triangle:
1.5^2 = x^2 + 1.5^2 - 2x(1.5)cosØ
x^2 = 3x cosØ
x = 3cosØ

total time = T = a/24 + x/3
T = (1/8)Ø + cosØ
dT/dØ = 1/8 - sinØ
= 0 for a min of T
sinØ = 1/8
Ø = .13533
so the arc he runs = 3Ø = .376 km

Answer 2

Since his swimming speed is so much slower than his walking speed, I am surprised that the angle is so small.

also, 3*0.13533 = 0.406

So, I suggest that Ø = π-arcsin(1/8) = 3.01626

The graph seems to confirm this:

http://www.wolframalpha.com/input/?i=x%2F8%2Bcosx

Answer 3

i know its late but for anyone looking at this now dont forget to check the case where they jog the whole distance and where they swim the whole distance

Answer 4

Total time T= theta/8 + cos(theta)
Put theta = π/2
You will get the least time 0.196 hours.