At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing
west at 25 knots and ship B is sailing north at 16 knots.
How fast (in knots) is the distance between the ships changing at 7 PM? (Note: 1 knot is a
speed of 1 nautical mile per hour.)
This and the other seem to be similar. Best to talk about the general method.
1. Find an expression for the distance between the two at time t.
In this case, with Pythagoras' help, that's
sqrt( (25t+30)^2 + (16t)^2)
= sqrt(881t^2 + 1500t + 900)
Nasty-looking thing. Call it
u = 881t^2 + 1500t + 900
and differentiate it.
du/dt = 1762t+1500
dy/du = 1/2sqrt(881t^2 + 1500t + 900)
so the whole thing is
(1762t+1500)/2sqrt(881t^2 + 1500t + 900)
Plug in t=7 and you're there.
=
To find the rate of change of the distance between the ships at 7 PM, we need to find the rates at which both ships are moving at that time.
Ship A:
Ship A is sailing west at a speed of 25 knots. Since the speed is given in knots, we can assume this is the speed in nautical miles per hour. So, Ship A is moving at 25 nautical miles per hour towards the west.
Ship B:
Ship B is sailing north at a speed of 16 knots. We assume this is also the speed in nautical miles per hour, so Ship B is moving at 16 nautical miles per hour towards the north.
Now, we can calculate the velocity vector for each ship:
The velocity vector for Ship A is (-25, 0) since it is moving west and not north.
The velocity vector for Ship B is (0, 16) since it is moving north and not west.
To find the relative velocity between the two ships, we subtract the velocity vector of Ship B from the velocity vector of Ship A:
Relative velocity vector = Velocity vector of Ship A - Velocity vector of Ship B
Relative velocity vector = (-25, 0) - (0, 16)
Relative velocity vector = (-25, -16)
Now, we know that the rate of change of the distance between two objects is equal to the magnitude of the relative velocity vector. So, to find the rate at which the distance between the ships is changing, we need to find the magnitude of the relative velocity vector.
Magnitude of the relative velocity vector = √((-25)^2 + (-16)^2)
Magnitude of the relative velocity vector = √(625 + 256)
Magnitude of the relative velocity vector = √881
Therefore, the rate at which the distance between the ships is changing at 7 PM is approximately equal to √881 knots.