Two boats leave the same port at the same time, with boat a traveling north at 15 knots (nautical miles per hour) and boat b traveling east at 20 knots. how fast is the distance between them changing when boat a is 30 nautical miles from port?

The chance in distance is 25 knots

Explanation:

The distance between the two particles is given by:

 (1)

Since A is traveling north and B is traveling east we can say that their displacement vector are perpendicular and therefore (1) transformed as:

(2)

Taking the differential with respect to time:

 (3)

where and are the respective given velocities of the boats. To find and we make use of the given position for A, , the Pythagoras theorem and the relation between distance and velocity for a movement with constant velocity.

with this time, we know can now calculate the distance at which B is:

and applying Pythagoras:

Now substituting all the values in (3) and solving for   we get: