Question

At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 kmyh and ship B is sailing north at 25 kmyh. How fast is the distance between the ships changing at 4:00 pm

Answer 1

Answer:

the distance between the ships is changing at 21.4 km/h

Step-by-step explanation:

Given;

distance between ship A and ship B = 150 km

speed of ship, A = 35 km/h

speed of ship B = 25 km/h

at 4 pm, the time difference = 4 hours

let the distance between A and B = C

The position of A after 4 hours = 35 km/h x 4 h = 140 km

The distance covered by A, a = 150 km - 140 km = 10 km

The distance covered by B, b = 25 km/h x 4 h = 100 km

The distance between A and B;

c² = a² + b²

c² = 10²  +  100²

c² = 10100

c = √10100

c = 100.5 km

The change in the distance between A and B is calculated as;

$c^2 = a^2 + b^2\\\\2c\frac{dc}{dt} = 2a\frac{da}{dt} + 2b\frac{db}{dt} \\\\c\frac{dc}{dt} = a\frac{da}{dt} + b\frac{db}{dt} \\\\100.5(\frac{dc}{dt}) = -10(35) + 100(25) \\\\(the \ negative \ sign \ indicates \ decrease \ in \ distance \ of \ A \ from \ B \ with \ time)\\\\100.5(\frac{dc}{dt})= 2150\\\\\frac{dc}{dt} = \frac{2150}{100.5} \\\\\frac{dc}{dt} = 21.39 \ km/h\\\\\frac{dc}{dt} \approx 21.4 \ km/h$

Therefore, the distance between the ships is changing at 21.4 km/h