Question

Sketch the situation if necessary and use related rates to solve. Two airplanes are flying in the air at the same height. Airplane A is flying east at 250 mph and airplane B is flying north at 300 mph. If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B, at what rate (in mph) is the distance between the airplanes changing?

Answer 1

-390 mph

Explanation:

Let a and b represent, respectively, the distances of A and B from the airport. The distance d between the planes is then given by the Pythagorean theorem as ...

d² = a² + b²

Differentiating with respect to time, we have ...

2d·d' = 2a·a' +2b·b'

Solving for d', we get ...

d' = (a/d)a' +(b/d)b'

The value of d at the time of interest is ...

d = √(a² +b²) = √(30² +40²) = √2500 = 50

Then the rate of change of separation is ...

d' = (30/50)(-250 mph) +(40/50)(-300 mph) = (-150 -240) mph

d' = -390 mph

The distance between planes is decreasing at 390 miles per hour.