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
Answer:
the distance between the cars is increasing at 40 mi / h four hours later
