Question 1196361
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Part (a)


Let's label the planes as PlaneA and PlaneB


PlaneA = northbound at 200 mph
PlaneB = southbound at 150 mph



Place planeA at (0,0) on the xy grid.
PlaneB will go to (80,0) which is 80 units away, to represent the 80 mile gap.


t = number of hours that elapse, after the planes are side by side


The location of planeA is (0,200t) since it travels 200 mph and it's heading north, i.e. along the positive y axis.
I'm using the idea that distance = rate*time
For instance, after t = 1 hour, plane A is at location (0,200t) = (0,200*1) = (0,200)


Meanwhile, the location of planeB is (80,-150t)
The negative is to indicate going south


20 min = 20/60 = 1/3 of an hour


Let's plug in t = 1/3 to find the location of each plane
A = (0,200t) = (0,200*(1/3)) = (0,200/3)
B = (80,-150t) = (80,-150*(1/3)) = (80,-50)


Now compute the distance from A to B
d = sqrt( (x1-x2)^2 + (y1-y2)^2 )
d = sqrt( (0-80)^2 + (200/3-(-50))^2 )
d = sqrt( (0-80)^2 + (200/3+50)^2 )
d = sqrt( 20,011.1111111111 )
d = 141.46063449282
d = 141.4606



Answer: Approximately 141.4606 miles
Round this however your teacher instructs.


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Part (b)


Let's go back to 
A = (0,200t)
B = (80,-150t)
and compute the distance based on these coordinates


d = sqrt( (x1-x2)^2 + (y1-y2)^2 )
d = sqrt( (0-80)^2 + (200t-(-150t))^2 )
d = sqrt( 6400 + (200t+150t)^2 )
d = sqrt( 6400 + (350t)^2 )
d = sqrt( 6400 + 122,500t^2 )


As a slight detour, let's plug in t = 1/3 to get...
d = sqrt( 6400 + 122,500t^2 )
d = sqrt( 6400 + 122,500(1/3)^2 )
d = 141.46063449282
We get the same result as before, so it helps confirm we have the correct general distance equation.


What we'll do from here is plug in d = 300 and solve for t
d = sqrt( 6400 + 122,500t^2 )
300 = sqrt( 6400 + 122,500t^2 )
300^2 = 6400 + 122,500t^2
90,000 = 6400 + 122,500t^2
6400 + 122,500t^2 = 90,000
122,500t^2 = 90,000
122,500t^2 = 90,000 - 6400
122,500t^2 = 83,600
t^2 = (83,600)/(122,500)
t^2 = 0.68244897959183
t = sqrt(0.68244897959183)
t = 0.82610470256006


The planes are 300 miles apart at the time marker of approximately 0.82610470256006 hours


Multiply by 60 to convert to minutes
60*0.82610470256006 = 49.5662821536037


Answer: Approximately 49.5663 minutes
Round this value however your teacher instructs.
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