Rob and Jill are running straight paths with constant veloci

Rob and Jill are running straight paths with constant velocity. We know Rob is at (-5,-4) at 12:00pm, and at (7, 5) at 12:10pm. Jill starts at (-4, 5) at 12:00pm, and at (3,-6) at 12:10pm. Show a table of values for every 5 seconds after 12:00. Also determine what parametric equations would represent each runner’s progress.

Solution

Rob is at (-5,-4) at 12:00pm, and at (7, 5) at 12:10pm.

We need after every 5 seconds....

Clearly Rob will be at the midpoint of -5,-4 and 7,5 at 12:05 pm

This would be at (1,0.5) at 12:05 pm

Consider (-5,-4) and (1,0.5) and parameterize...
(-5,-4) + t((1,0.5) - (-5,-4))
(-5,-4) + t(6,4.5)
(-5 + 6t , -4 + 4.5t)

So, the parametric equation is :
x = -5 + 6t
y = -4 + 4.5t
where t is every 5 minute duration with t = 0 representing 12:00 pm

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Now, Jill :
Jill starts at (-4, 5) at 12:00pm, and at (3,-6) at 12:10pm

Midpoint is (-0.5,-0.5)

So, (-4,5) and (-0.5,-0.5)

r = (-4,5) + t((-0.5,-0.5) - (-4,5))

r = (-4,5) + t(3.5 , -5.5)

r = <-4 + 3.5t , 5 - 5.5t>

So, x = -4 + 3.5t
and y = 5 - 5.5t
where t represents every 5 minute duration with t = 0
representing 12:00 pm