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 17 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

To find the speed at which the distance between the ships is changing, we need to calculate the rate of change of the distance.

First, let's consider the positions of the ships at 5 PM. Since ship A sailed west at a constant speed of 16 knots for 5 hours (from noon to 5 PM), it would have traveled a distance of 16 * 5 = 80 nautical miles to the west.

So ship A would be 50 + 80 = 130 nautical miles due west of ship B at 5 PM.

To calculate the distance between the ships, we can use the Pythagorean theorem. The distance between the ships is the hypotenuse of the right-angled triangle formed by the positions of the ships.

The horizontal distance between the ships (x-coordinate) is the westward distance that ship A has traveled, which is 80 nautical miles. The vertical distance between the ships (y-coordinate) is the northward distance that ship B has traveled, which is 17 knots * 5 hours = 85 nautical miles.

Using the Pythagorean theorem, we find that the distance between the ships is √(80^2 + 85^2) ≈ 116.12 nautical miles.

To find the rate of change of this distance, we need to find the derivative of the distance function with respect to time. Differentiating the distance equation gives:

d(distance) / d(time) = [d/d(time)] (√(80^2 + 85^2))

Using the chain rule, we can differentiate this equation. Taking the derivative of the square root function gives:

d(distance) / d(time) = 1 / (2 * √(80^2 + 85^2)) * [d/d(time)] (80^2 + 85^2)

The derivative of (80^2 + 85^2) with respect to time is 0, since the values of 80 and 85 are constants.

Simplifying the equation:

d(distance) / d(time) = 1 / (2 * √(80^2 + 85^2)) * 0

Since the derivative of the expression (80^2 + 85^2) with respect to time is zero, the rate of change of the distance between the ships at 5 PM is 0 knots.