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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?

 

  • Amy and Allison do not collide.
  • Amy and Allison collide at 3.6 seconds.
  • Amy and Allison collide at 2.4 seconds.
  • Amy and Allison collide at 4.8 seconds.
 May 9, 2020
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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!

 May 9, 2020

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