#### To determine

**To find:** The rate at which the people moving apart 15 min after the women starts walking.

#### Answer

The required rate at which the people moving apart is dzdt=8378,874≈8.99 ft/s.

#### Explanation

**Given:**

A man starts walking towards North at 4 ft/s from a fixed point P.

After 5 minutes later a women starts walking towards South at 5 ft/s from a 500 ft due East of the fixed point P.

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Pythagorean Theorem for the Right angle triangle.

**Calculation:**

Let *P* be the fixed point and x is the distance travelled by man towards North, and z is the distance between the people and y is the distance travelled by the women from 500 ft due to the point *P* towards South as shown below in the Figure 1.

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

Hence x, y and z are the function of the time variable t .

Apply Pythagorean Theorem in the above triangle ABC,

z2=(x+y)2+5002

Obtain dzdt, 15 minutes after the woman starts walking.

After 15 minutes later the position of man from the point P:

x=velocity×time=4×20×60 [Q 1 min=60 s]=4,800 ft

After 15 minutes later position of y from the point P:

y=velocity×time=6×15×60 [Q 1min=60 s]=4,50 0 ft

And distance between man and woman 15 minutes later:

z2=[(4,800+4,500)2+5002]=[9,3002+5002]=[86,490,000+250,000]=[86,740,000]

On further simplification the required distance between people.

z=86,740,000 ft

Differentiate z with respect to the time t.

ddt[z2]=ddt[(x+y)2+5002]2zdzdt =2(x+y)[dxdt+dydt]dzdt =(x+y)z[dxdt+dydt] where z≠0

Substitutes 4,800 for x and 4,500 for y and 4 for dxdt and 5 for dydt and 86,740,000 for z in dzdt.

dzdt=(4,800+4,500)86,740,000[4+5]=9,3001008,674[9]=93×98,674=8378,674 ft/s

On further simplification the required rate of change of distance between the people is as follows.

dzdt=8378,674≈8.99 ft/s

Therefore, the required rate at which the people moving apart is dzdt=8378,874≈8.99 ft/s.