Question

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?

Answer 1

The girls don't collide

Step-by-step explanation:

Amy and Allison will collide if there is a value of t on the interval [0, 50] that makes the x and y position equal.

The equations for Amy are

x(t) = 42+(2/3)t

y(t) = 8+(1/6)t

The equations for Allison are

x(t) = 12 + t

y(t) = 5 + (1/4)t

To find the value of t, we will make the expressions for x(t) equal, so

42+(2/3)t = 12 + t

Solving for t, we get

42 + (2/3)t - (2/3)t = 12 + t - (2/3)t

42 = 12 + (1/3)t

42 - 12 = 12 + (1/3)t - 12

30 = (1/3)t

3(30) = 3(1/3)t

90 = t

It means that after 90 tenths of seconds, the x coordinate of Amy and Allison will be the same. However, t is on the interval [0, 50] and 90 doesn't belong to that interval. It means that the girls don't collide.