precalc question

Question

Marla is running clockwise around a circular track. She runs at a constant speed of 3 meters per second. She takes 46 seconds to complete one lap of the track. From her starting point, it takes her 12 seconds to reach the northermost point of the track.

(a) Give Marla’s coordinates at her starting point.

(b) Give Marla’s coordinates when she has been running for 10 seconds.

(c) Give Marla’s coordinates when she has been running for 901.3 seconds.

Answer 1

Mara,

first associate an angle with each time lapse:

time lapse angle covered distance covered(arclength)

46 sec 360degrees 46sec*(3m/sec)=138meters

12 sec (xdegress/12)=(360degrees/46sec)

xdegrees=93.3degrees 12*3=36meters

10 sec (xdegress/10)=(360degrees/46sec)

xdegrees=78.3degrees 10*3=30meters

901.3sec xdegrees =7053.7 degrees 2703meters and notice this is short of 20 laps (2760m).

Next compute the radius from the complete circle: circumference 138m----->radius r=138/(2pi) = 22meters

Now place the running track in a coordinate system. Center at the origin, North point (0,22).

12second problem:

SInce the rotation is clockwise and after 12 seconds the runner is at the northernmost point (top), the runner started in quadrant III, because the runner covers 93.3degrees in 12 sec.

Coordinates x=r cos theta, y=rsin theta. But theta =93.3+90 degrees. (You have to measure the angle from the positive x-axis.) ---> coordinate point (x=22cos183.3 ,y=22sin183.3)

10second problem:

covers 78.3degrees clockwise from starting point (land in quadrant II), how many degrees short of northern most point: 15 degrees-----> New angle from positive x-axis: 90+15 degrees.

position after 10 sec: (x=22cos105, y=22sin105)

901.3 sec problem.

covers 7053.3degrees from starting point=19*360+213.3. The runner covered 19 complete circles(back at starting point) and an additional 213.3 degrees. What is the terminal angle of these many degrees?

It is 93.3 degrees to get to the north point. So 213.3-93.3=120 degrees from the north point clockwise. You get to -30degrees (terminal from x-axis):

coordinates (x=22cos(-30), y=22sin (-30)).