To find how fast the distance between the ships is changing at 3 PM, we will need to use the concept of rates of change.
Let's start by drawing a diagram to represent the situation. We can assume a Cartesian coordinate system, with the origin (0,0) being the initial position of ship B.
At noon, ship A is 10 nautical miles due west of ship B. This means ship A is located at (-10,0). Ship A is sailing west at 15 knots which implies that its coordinates are changing with time at a rate of (-15, 0) nautical miles per hour (westward). Ship B is sailing north at 15 knots, resulting in a rate of (0, 15) nautical miles per hour (northward).
We want to find the rate at which the distance between the ships is changing at 3 PM. Let's assume the distance between the two ships at the given time be represented by d(t).
To find d(t), we can use the distance formula between two points in a Cartesian coordinate system:
d(t) = √[(x_A - x_B)^2 + (y_A - y_B)^2]
where (x_A, y_A) represents the coordinates of ship A, and (x_B, y_B) represents the coordinates of ship B.
Using the given information, at 3 PM ship A will have traveled 3 hours from noon, while ship B will have traveled 3 hours northward.
The new coordinates of ship A would be:
x_A = -10 - (15 * 3)
y_A = 0
The new coordinates of ship B would be:
x_B = 0
y_B = 15 * 3
Now, we can calculate the distance d(t) at 3 PM by substituting these values into the distance formula:
d(t) = √[(-10 - (15 * 3))^2 + (0 - (15 * 3))^2]
Simplifying the above expression will give us the distance between the two ships at 3 PM.
Finally, we can differentiate the equation with respect to time 't' to find the rate at which the distance is changing:
d(d(t))/dt = d'(t)
This will give us the rate at which the distance between the ships is changing at 3 PM.