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?

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

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

Total time T= theta/8 + cos(theta)

Put theta = π/2
You will get the least time 0.196 hours.

To find the distance Peter should jog in order to arrive at point W in the least amount of time, we need to consider the time it takes for him to jog and the time it takes for him to swim.

First, let's find the time it takes for Peter to jog along the north shore. Since he is jogging at a rate of 24 km/h, the time it takes can be calculated using the formula:

Time = Distance / Speed

The distance he needs to jog is half the circumference of the circle, as he starts at one point on the shore and needs to reach the diametrically opposite point. The formula for the circumference of a circle is:

Circumference = π * Diameter

Given that the diameter is 3 km, the distance Peter needs to jog is:

Distance_jog = 0.5 * π * Diameter

Substituting the given values:

Distance_jog = 0.5 * π * 3 km

Now we can calculate the time it takes for Peter to jog:

Time_jog = Distance_jog / Speed_jog

Substituting the given jogging speed:

Time_jog = (0.5 * π * 3 km) / 24 km/h

Next, let's find the time it takes for Peter to swim the straight line distance to W. The distance he needs to swim is the diameter of the circle, which is 3 km. We already know his swimming speed is 3 km/h.

Time_swim = Distance_swim / Speed_swim

Substituting the given values:

Time_swim = 3 km / 3 km/h

Now, we can calculate the total time it takes for Peter to reach point W:

Total_time = Time_jog + Time_swim

Substituting the calculated values, we can find the distance Peter should jog to minimize the total time.

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