Amy and Allison are both running toward the soccer ball during a game. Amy's path is represented by the parametric equations x(t)=42+2/3t,y(t)=8+1/6t, where t is on the interval [0,50] and t is measured in tenths of seconds. Allison's path is represented by the parametric equations x(t)=12+t,y(t)=5+1/4t, where t is on the interval [0,50] and t is measured in tenths of seconds. Do the girls collide? If so, when do they collide?
+23246 I'm going to change each of these into rectangular coordinates.
x = 42 + (2/3)t ---> solve for t ---> x - 42 = (2/3)t ---> 3x - 126 = 2t ---> t = (3x - 126)/2
y = 8 + (1/6)t ---> substitute ---> y = 8 + (1/6)(3x - 126)/2 ---> y = (1/4)x - 15/6
x = 12 + t ---> solve for t ---> t = x - 12
y = 5 + (1/4)t ---> substitute ---> y = 5 + (1/4)(x - 12) ---> y = (1/4)x + 2
Since they have the same slope, they don't intersect -- they do not collide!
