# calculus

two trains are on parallel tracks that are 2 miles apart. If both trains are traveling at 40 mph, how fast is the distance between the trains decreasing when they are 8 miles apart?

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1. Hmmm. They are coming toward each other , distance decreasing, and you want to know when the are 8 miles apart when they started 2 miles apart? Makes no sense.

relative velocity=80mph
that is how fast the distance between them is decreasing at any time. This problem is not thought out, or typed wrong. Perhaps it would make better sense if the word "parallel" were change to perpendicular, however, it is very difficult to start them at 2mi apart, move them toward a common point, and sometime later be 8 mi apart.

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2. The parallel tracks are 2 miles apart

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3. If one train is x miles from the point of closest approach, and the other is y, then

dx/dt = dy/dt = -40

the distance z between the trains is

z^2 = (x+y)^2 + 2^2
So, when they are 8 miles apart,
(x+y)^2 + 4 = 64
x+y = √60

z dz/dt = (x+y)(dx/dt+dy/dt)
8 dz/dt = √60(-40-40)
dz/dt = -10√60
= -77.5

Sounds reasonable. If they were on the same track, they'd be approaching at 80 mph, so we'd expect the speed to be a little less than that.

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