#### To determine

**To find:** The rate of change of the distance between the ships.

#### Answer

The rate of change of the distance between the ships is dzdt=72013≈55.40 km/h .

#### Explanation

**Given:**

Ship A is sailing South at 35 km/h, that is dxdt=35 km/h.

Ship B is sailing North at 25 km/h, that is dydt=25 km/h.

**Formula used:**

Chain rule: dydx=dydu⋅dudx

**Calculation:**

Let x be the distance travelled towards South by the ship A till 4 pm, and y be the distance travelled towards North by the ship B till 4 pm, and z be the distance between the ships at 4 pm as shown in the Figure-1 given below.

Since x, y and z increases as time t increases, x, y and z are the function of the time t.

Obtain dzdt.

Obtain *z* by using Pythagorean Theorem in above triangle.

z=(x+y)2+1002=(x+y)2+1002

The distance travelled by the ship A till 4 pm,

x=35×4=140 km

The distance travelled by the ship B till 4 pm,

y=25×4=100 km

The distance between the ships at 4 pm,

z=(140+100)2+1002=2402+1002=57600+10000=67600 km

Differentiate z with respect to the time t,

ddt[z2]=ddt[(x+y)2+1002]2zdzdt=2(x+y)[dxdt+dydt]dzdt=(x+y)z[dxdt+dydt]

Substitute x=140, y=100, z=67600 and dxdt=35, and dydt=25 in dzdt.

dzdt=(140+100)67600[35+25]=24067600×60=240260×60=24×3013

That is, dzdt=72013≈55.40 km/h.

Therefore, the rate of change of the distance between the ships is 55.40 km/h .