Step 1

t = time,

y = position of the first car travelling north,

x = position of the second car travelling east,

z = distance between the two cars.

In the diagram in Figure 3.2.3, the point P is placed at the origin.

Figure 3:2.3

Step 2

x² + y² = z².

Step 3

, whence

Step 4

We first find the values of x, y, and z at the time t = 2 hrs. We are given that when t = 1, y = 0. In the next hour the car goes 40 miles, so at t = 2, y = 40. We are given that at time t = 2½, x = 0. One-half hour before that the car was 30 miles to the left of P, so at t = 2, x = -30. To sum up,

at t = 2, y = 40 and x = - 30.

We can now find the value of z at t = 2,

z = = 50.

Finally, we solve for dz/dt at t = 2,

60 · (-30) + 40 · 40 = , -4.

The negative sign shows that z is decreasing. Therefore at 2:00 the cars are getting closer to each other at the rate of 4 mph.