#### To determine

**To find:** At what rate the distance between the cars increasing two hours later.

#### Answer

The required rate at which the distance between the cars increasing is dzdt=65 mi/h.

#### Explanation

**Given:**

Two cars start moving from the same point.

One car travel towards South at 60 mi/h and another travel towards West at 25 mi/h.

That is, dxdt=60 mi/h and dydt=25 mi/h.

**Formula used:**

Pythagorean Theorem

**Calculation:**

Let x is the distance travelled by first car toward West and reach to the point C and y is the distance travelled by second car toward South and reach to the point B as shown in the Figure 1 given below.

The position of both cars is shown in the given below Figure-1 after two hours later.

Let z be the distance between the cars after two hours later as shown the above figure-1.

Then by using the Pythagorean Theorem

z2=x2+y2

Since x and y both changes with the time t, z also changes with time.

Obtain the position of cars two hours later:

Distance travelled by the first car toward West direction is

x=2(60)=120

Distance travelled by the second car toward South direction is:

y=2(25)=50

Therefore, distance between the both cars two hours later is

z=(120)2+(50)2=14,400+2,500=16,900=130

Differentiate z2=x2+y2 with respect to the time t .

ddt(z2)=ddt(x2+y2)2zdzdt =2xdxdt+2ydydtdzdt =1z[xdxdt+ydydt]

Substitute 120 for x and 50 for y and 60 for dxdt and 25 for dydt and 130 for z in dzdt

dzdt=1130[(120)(60)+(50)(25)]=1130[7,200+1,250]=1130[8,450]=65 mi/h

Therefore, the required rate at which the distance between the cars increasing is dzdt=65 mi/h.