Sandra can run a mile in 6 minutes. At her high school track, shown below, 4 laps around the track equals 1 mile. If Sandra starts at point S, which parametric equations will find her position on the track at any time t in seconds?
we know that 1) Sandra can run a mile in 6 minutes-------> 6*60-----> 360 sec 2) 4 laps around the track equals 1 mile so 4 laps around the track in 360 sec 1 lap in 360/4--------> 90 sec
3) the position of Sandra for t=90 sec must be equal to the point S (0,56) I proceed to analyze each case for t=90 sec
case a) x(t)=-140 cos(pi*t/45) y(t)=112 sin(pi*t/45) x(t)=-140 cos(pi*90/45)------> -140 y(t)=112 sin(pi*90/45)-------> 0 the position is the point (-140,0)------> is not the point S case b) x(t)=140 sin(pi*t/90) y(t)=-112 cos(pi*t/90) x(t)=140 sin(pi*90/90)------> 0 y(t)=-112 cos(pi*90/90)-------> 112 the position is the point (0,112)------> is not the point S case c) x(t)=-70 sin(pi*t/45) y(t)=56 cos(pi*t/45) x(t)=-70 sin(pi*90/45)------> 0 y(t)=56 cos(pi*90/45)
-------> 56 the position is the point (0,56)------> is equal to the point S----> is the solution case d) x(t)=70 cos(pi*t/90) y(t)=-56 sin(pi*t/90) x(t)=70 cos(pi*90/90)------> -70 y(t)=-56 sin(pi*90/90)-------> 0 the position is the point (-70,0)------> is not the point S
therefore the answer is the option C x(t)=-70 sin(pi*t/45) y(t)=56 cos(pi*t/45)
