Question

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 40 seconds to run a lap of the track. George and Paula pass each other after 13 seconds.
After running for 4 minutes, how far east of his starting point is George? (Round your answer to three decimal places.)

Answer 1

yes that's an accurate picture I know the equation in x=rcos(pheta-omega(t)) but I need help finding the radius and pheta

Answer 2

so to find the radius we will need to use these three statements:
"George runs at 8 feet per second.
"Paula takes 40 seconds to run a lap of the track.
"George and Paula pass each other after 13 seconds.

george runs 104 feet in 13 seconds, becuase 13seconds*(8ft/sec)

if they run towards each other from the opposite sides, then they are only having to run half the track, so that distance is half the circumfrence thus = pi*r

so,
pi*r = 104 + 13seconds*x
where x is paula's runnig speed. thus we need to know paula's speed

paula takes 40seconds for the whole track, the whole track is the circumfrence = 2pi*r,
so,
2pi*r = 40seconds * (x ft/sec)
where x is the speed paula can run at.
rearrange the equation:
(2pi*r)/40=x

take this equation just made and this one we made before:
=x
pi*r = 104 + 13seconds*x
it becomes:
pi*r = 104 + 13*((2pi*r)/40)
so now you can solve for r

Answer 3

ok so I got 2080/7pi for the radius

Answer 4

correct

Answer 5

can you get the rest? ya no?

Answer 6

Could you walk me through it?

Answer 7

so now we know the length of the track is 2*pi*r = 594.29 feet.

we next need to figure out how many times George ran around the track in 4 minutes.
we know George runs 8ft per second, and ran for 4 minutes and that the track is 594.29 feet per lap. with all this we can obtain how many laps George ran around the track.

so geoge ran araound the track 3.23 times. so he passes the point that he starts at ( the far west point on the track) 3 times. and continues 0.23 laps more.

so we want to know how far 0.23 laps are in feet.

137.13 feet is how far george has ran past the starting line for George.


we now want to know what angle, pheta, george runs through.