Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 427 knots (nautical miles per hour; a nautical mile is 2000 yd or 6000 ft.) Plane B is approaching the intersection at 439 knots.

At what rate is the distance between the planes decreasing when Plane A is 5 nautical miles from the intersection point and Plane B is 5 nautical miles from the intersection point?

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To find the rate at which the distance between the planes is decreasing, we need to calculate the derivative of the distance between the planes with respect to time.

Let's assume that Plane A is moving along the x-axis and Plane B is moving along the y-axis. The distance between the planes is then given by the Pythagorean theorem:

d^2 = x^2 + y^2

where d is the distance between the planes, and x and y are the distances of Plane A and Plane B from the intersection point, respectively.

Taking the derivative of this equation with respect to time t, we get:

2d * (dd/dt) = 2x * (dx/dt) + 2y * (dy/dt)

Simplifying this equation, we get:

dd/dt = (x * (dx/dt) + y * (dy/dt)) / d

Since we are given the distances of Plane A and Plane B from the intersection point (x = 5 nautical miles and y = 5 nautical miles) and the speeds of Plane A and Plane B (dx/dt = 427 knots and dy/dt = 439 knots), we can substitute these values into the equation to find the rate at which the distance between the planes is decreasing.

To find the rate at which the distance between the planes is decreasing, we need to determine the rate at which Plane A is approaching Plane B. We can then use this information to calculate the rate of decrease in distance.

Let's gather the given information:
- Plane A is flying at 427 knots.
- Plane B is flying at 439 knots.
- Both planes are 5 nautical miles away from the intersection.

To solve this problem, we can use the concept of relative velocity. The relative velocity between the two planes is the difference in their velocities. In this case, the relative velocity is given by:

Relative Velocity = 439 knots - 427 knots

Next, we need to convert the relative velocity to feet per hour since the altitude is given in feet. We know that 1 knot is equal to 6000 feet per hour since a nautical mile is 6000 feet. Therefore, we can calculate:

Relative Velocity in feet per hour = (439 knots - 427 knots) * 6000 feet per hour

Now, we have the relative velocity between the planes. The final step is to calculate the rate of decrease in distance. To do this, we can use the following formula:

Rate of decrease in distance = Relative velocity / (distance between the planes)

In this case, the distance between the planes is constant at 5 nautical miles. However, we need to convert this distance to feet to match the units of the relative velocity. Since 1 nautical mile is equal to 6000 feet, we can calculate:

Rate of decrease in distance = Relative velocity / (5 nautical miles * 6000 feet per nautical mile)

Simply substitute the values into the equation and perform the calculations to find the rate at which the distance between the planes is decreasing.