Question

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 400 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

It is given that airplane A is flying east at a speed of 250 mph towards the airport located 30 miles east and airplane B is flying north at a speed of 400 mph towards the same aiirport located 40 miles north.  And this can be represenred as follows

From th figure we can find that the distance between the two airplanes is AB which is the hypotunuse of th triangle ABC.

Let

Since both the airplanes are moving then both the sides AC and BC of the triangle are also changing and the rate at which these sides are changing is equal to the rate at which the airplannes are moving hence mathematically we can write

Since the distance AC and BC are decreasing as the airplanes are approaching towards the airport

On applying Pythagorous Theorem which is given by

Since

  

We need to find the rate at which the distance between the airplanes are changing, and this is equal to the rate at which th hypotunuse is changing because hypotunuse represents the distance between the airplanes so it is required to find

We know

Taking the derivative with respect to time

Substituting the values

  

  

Hence the rate at which the distance between the two airplanes is