Two trains leave a station, one headed due west and the other headed due south. How fast is the southern bound train moving if the west bound train is 15 miles from the station traveling at a constant speed of 35 mph, the distance between the trains is 24 miles, and the distance between them is increasing at a rate of 25 mph.

dx/dt = 4 mph

Step-by-step explanation:

The two trains, with the station ( origin of the coordinates system) all, form a right triangle when we considered the distance between the two trains.

In that triangle:

distance between trains  D = 24 miles

leg (1 ) distance of train going west to origin  y = 15 miles

leg (2) distance of train going south x = ??

When d = 24 and y = 15

D² = y² + x²      ⇒   x² = D² - y²

x = √ (24)² - (15)²

x = √351

x = 18,73 miles

Now according to Pythagoras theorem

D² = x² + y²

Taking derivatives (respect to time) on both sides of the equation we get

2*D*dD/dt = 2*y*dy/dt  + 2*x*dx/dt

In this expression we are looking for dx/dt, all the rest of variables are known

2*24*25 = 2* 15*35  +  2*(18,73)* dx/dt

(1200 - 1050 )/37,46 = dx/dt

dx/dt = 4 mph