Precalc question

Question

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost point. They both run counterclockwise. Alexandra runs at 4 meters per second, and will take exactly 2 minutes to catch up to Charlie. Impose a coordinate system, and give the x- and y-coordinates of Charlie after one minute of running.

Answer's (59.84016, 4.35666), but idk how to solve it. Thanks to whoever solves this.

Answer 1

To start we need to figure out how fast Charlie is running so we can calculate his position after one minute. He starts with a lead of 1/4th of the track. The circumference of the track is 120pi so his lead is 30pi. Since Alexandra catches up to him in 4 seconds at that point they will have traveled the same distance. Alexandra travels 4M/sec for 120 sec= 480M. Charlie travels xM/sec for 120 seconds plus his lead of 30pi

So 30pi + 120x = 480. Solving for x = 3.2146 seconds

In 60 seconds Charlie travels 60(3.2146) = 192.876 M

From Charlies initial position which is half way around the track if he were to travel back to a position at the easternmost point ( this would be at 0 degrees) he would have to travel the distance of 1/2 the circle which is 60pi= 188.495M so his position is an arc length of 192.876 - 188.495 = 4.38M past zero.

Now we need to calculate the central angle subtended by this arc which is what percent of the entire circumference x. 360 degrees

(4.38/120pi) x 360 = 4.183 degrees.

We can calculate the x and y coordinates with some right angle trig. The central angle is 4.183 degrees and the hypotenuse is the radius of the track( 60)

Cosine 4.183 = x/60 x= 59.84

Sine 4.183 = y/60 y= 4.36

Answer 2

In circular motion, the linear velocity (v) equals the angular velocity (ω) multiplied by the radius (r) so v = rω or ω = v/r. Using that equation, we can calculate the angular velocity of Alexandra to be 4/60 = 1/15 radians/second.

Also in circular motion, the angle (θ) = the angular velocity (ω) multiplied by the time (t) so θ = ωt. Using that equation, the angle that Alexandra ran is (1/15 radians/second)(120 seconds) = 8 radians. There are 2π radians in one complete circle so that puts her all the way around plus another quarter or so.The angle she ran past the location that Charlie started is 8 - 2π - π/2 = 0.14602 radians. Charlie ran a total of 2π + 0.14602 radians = 6.4292 radians and he did so in 120 seconds so his angular velocity was 6.4292/120 = 0.053577 radians/sec.

So after 1 minute, Charlie ran through θ = ωt or (0.053577)(60) = 3.2146 radians. That location is shown in the diagram as C₂.

To find the x and y coordinates of C₂, it will be easier to consider the angle in standard location. To do so, we need to subtract π radians. So, in standard position, the location is an angle of 3.2146 - π = 0.073 radians. We can use trig. to find the x and y coordinates: x = rcos(θ) = 60cos(0.073) = 59.84016 and y = rsinθ) = 60sin(0.073) = 4.37666