At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 3 PM

To find how fast the distance between the ships is changing, we need to use the concept of rate of change. In this case, we want to find the rate of change of the distance between the ships at 3 PM.

Let's break down the problem and find the distance between the ships at noon and 3 PM.

At noon:
- Ship A is 50 nautical miles due west of ship B.

From noon to 3 PM:
- Ship A is sailing west at 16 knots for 3 hours, covering a distance of 16 knots/hour × 3 hours = 48 nautical miles.
- Ship B is sailing north at 21 knots for 3 hours, covering a distance of 21 knots/hour × 3 hours = 63 nautical miles.

To find the distance between the ships at 3 PM, we can use the Pythagorean theorem since ship A has moved west and ship B has moved north. The distance between the ships is the hypotenuse of a right-angled triangle.

Distance between the ships at 3 PM = √(48^2 + 63^2) nautical miles

Now, to find how fast the distance between the ships is changing at 3 PM, we need to differentiate the distance equation with respect to time (t).

First, let's define some variables:
- Let x be the distance traveled by ship A (westward) at 3 PM.
- Let y be the distance traveled by ship B (northward) at 3 PM.
- Let d be the distance between the ships at 3 PM.

Using the chain rule of differentiation, we have:

d/dt [(x^2 + y^2)^0.5] = (1/2) * [(x^2 + y^2)^(-0.5)] * (2x*dx/dt + 2y*dy/dt)

Substituting the given values:
- x = 48 nautical miles
- y = 63 nautical miles

We also need to find dx/dt and dy/dt, i.e., the rates at which ship A and ship B are moving at 3 PM.

From the given information, ship A is moving at a constant rate of 16 knots/hour, and ship B is moving at a constant rate of 21 knots/hour. Therefore:

dx/dt = 16 knots/hour
dy/dt = 21 knots/hour

Now, substituting these values into the differentiation equation, we can find how fast the distance between the ships is changing at 3 PM.

d/dt [(48^2 + 63^2)^0.5] = (1/2) * [(48^2 + 63^2)^(-0.5)] * (2 * 48 * 16 + 2 * 63 * 21) nautical miles/hour

Evaluating this expression will give you the rate at which the distance between the ships is changing at 3 PM (in knots).