Step 1 :Given two trains, one moving west towards Denver at a speed of 2 mi/min and another moving north away from Denver at a speed of 1.5 mi/min. At time t=0, the first train is 20 mi east of Denver and the second train is 10 mi north of Denver.
Step 2 :We can use the Pythagorean theorem to find the distance between the two trains at any given time. The distance is a function of time, given by \(d = \sqrt{(20 - 2t)^2 + (1.5t + 10)^2}\).
Step 3 :To find the time when the trains are closest together, we take the derivative of the distance function and set it equal to zero. This gives us \(\frac{d}{dt} = \frac{6.25t - 25.0}{\sqrt{(20 - 2t)^2 + (1.5t + 10)^2}} = 0\).
Step 4 :Solving this equation gives us the time at which the trains are closest together, \(t = \boxed{4}\) minutes.
Step 5 :Substituting this time back into the distance function gives us the minimum distance between the two trains, \(d = \sqrt{(20 - 2*4)^2 + (1.5*4 + 10)^2} = \boxed{20}\) miles.