George and Paula are running around a circular track George

George and Paula are running around a circular track. George starts at the westernmost point of the track, and Paula starts at the easternmost point. The illustration below shows their starting positions and running directions. They start running toward each other at constant speeds. George runs at 8 feet per second. Paula takes 50 seconds to run a lap of the track. George and Paula pass each other after 11 seconds.

After running for 4 minutes, how far east of his starting point is George? (Round your answer to three decimal places.)

Solution

let the circle have radius r feet
the circumference, hence, is 2r feet
now george and paula pass each other after 11 seconds, so in that time george has covered 88 feet of distance
it takes paula 50 seconds to run a lap, and speed = distance / time
speed = 2r / 50
= r / 25 feet per second
so in 11 seconds she ran 11r / 25 feet
combined, they covered 88 + (11r / 25) feet. but combined, they covered the entire top half of the circle, which is equal to r
so 88 + (11r / 25) = r
2200+ 11r = 25r
2200 = 14r
2200/7 = 2r = circumference
r = 2200 / 44
r = 50 ft
now, after 4 minutes, george has travelled 8(4)(60) = 1920 feet
that means he runs 6 complete laps (1886 feet) and runs 34ft of the next one
arc length = rø, so ø = arc length / r = 34 / 50 = 0.68 rad
now consider a right angled triangle in the second quadrant with hypotenuse r and angle 0.68
so cosø = adj side / r
0.77 = adj side / 50
adj side = 38.88
so the total distance east, since he started from the westernmost point, is 50 - adj side = 11.121 ft