Question

At noon, ship A is 130 km west of ship B. Ship A is sailing east at 30 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 4:00 PM?

Answer 1

Answer:

36.1 km/h

Step-by-step explanation:

rate at which the distance is changing is given by relative velocity(vR)

Look at the diagram. THe diagram shows relative motion. From relative motion diagram, velocity vector diagram is drawn.

From velocity vector diagram, it can be seen that relative velocity or vR can be calculated using pythagoras theorem.

vR²= vA² + vB²

vR²= 30²+ 20²

vR= 36.1 km/h

Answer 2

Answer:

At 4:00pm, the distance between ship A and B is increasing at the rate of 16.12 km/hr

Step-by-step explanation:

The illustration forms a right angle triangle.  At noon the distance between ship A and Ship B is 130 km . But ship A is west of ship B . Ship A is sailing east towards the starting point of ship B which means the distance is shrinking  at 30 km/hr and ship B is sailing north at 20 km/hr .This means ship B distance is extending.

The distance covered by ship A is  

distance = speed × time

noon to 4 pm  =  4 hrs

distance = 30 × 4

distance = 120 km

The distance covered by ship B is  

distance = speed × time

distance = 20 × 4

distance = 80 km

The distance from Ship A present position to ship B after moving east is 130 km - 120 km = 10 km.

Using Pythagorean theorem

c = ?

a = 10 km

b = 80 km

c² = a² + b²

c² = 10² + 80²

c² = 100 + 6400

c² = 6500

c = √6500

The question ask us how fast is the distance between the ships changing at 4:00 pm. This means calculating the rate .

c² = a² + b²

differentiate with respect to time

2c(dc/dt) = 2a(da/dt) + 2b(db/dt)

divide through by 2

c(dc/dt) = a(da/dt) + b(db/dt)

since we are looking for the rate of c we make dc/dt subject of the formula

dc/dt = (a(da/dt) + b(db/dt)) / c

Note Ship A travelling to the east is shrinking. so the speed which is da/dt = - 30 km/hr

c = √6500

a = 10 km

b = 80 km

dc/dt =  10(-30) + 80(20) / √6500

dc/dt = (-300 + 1600)/√6500

dc/dt = 1300 /√6500

dc/dt = 1300/80.622577483

dc/dt = 16.1245154966 km / hr

At 4:00pm, the distance between ship A and B is increasing at the rate of 16.12 km/hr