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?

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.

The parallel tracks are 2 miles apart

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.

To find how fast the distance between the trains is decreasing, we need to use the concept of related rates. Let's break down the problem step by step:

1. Identify the given information:
- The distance between the trains is initially 2 miles.
- The trains are traveling at a speed of 40 mph each.

2. Define the variables:
- Let's call the distance between the trains at any time "d" (in miles).
- We'll use "t" to represent time (in hours).

3. Determine the rate at which the distance between the trains is changing:
- We are looking for d/dt (the rate of change of distance with respect to time).

4. Understand the relationship between the variables:
- Since the trains are traveling in parallel, the distance between them will decrease at a rate equal to the combined speed of the two trains.
- In this case, both trains are traveling at a speed of 40 mph, so the rate of change of distance (d/dt) will be twice that.

5. Set up the equation:
- We know that d/dt = 2 * 40 mph = 80 mph.

6. Find the rate at which the distance is decreasing when the distance between the trains is 8 miles:
- Plug in the known values into the equation: d/dt = 80 mph.
- Since the distance between the trains is currently 8 miles, the rate at which it is decreasing will also be 80 mph.

Therefore, the rate at which the distance between the trains is decreasing when they are 8 miles apart is 80 mph.