Two cars start moving from the same point. one travels south at 32 mi/h and the other travels west at 24 mi/h. at what rate is the distance between the cars increasing four hours later?

You can model the problem as a rectangle triangle.
 We define variables:
 a = Car travels south
 b = Car travels west
 c = distance between them

 After four hours we have the following distance values:
 a = (32) * (4) = 128
 b = (24) * (4) = 96
 By the Pythagorean theorem we have:
 c = root (a ^ 2 + b ^ 2)
 c = root ((128) ^ 2 + (96) ^ 2)
 c = 160
 On the other hand, we have that the speeds are:
 da / dt = 32
 db / dt = 24
 We must find the value of dc / dt.
 For this, we derive the equation of distance with respect to time.
 We have then:
 a ^ 2 + b ^ 2 = c ^ 2
 Deriving:
 2a da / dt + 2b db / dt = 2c dc / dt
 Substituting values:
 2 * (128) * (32) + 2 * (96) * (24) = 2 * (160) dc / dt
 Clearing we have:
 dc / dt = (2 * (128) * (32) + 2 * (96) * (24)) / (2 * (160))
 dc / dt = 40 mi / h
 
 the distance between the cars is increasing at 40 mi / h four hours later