# Two aircraft approach airfield at the same constant altitude.  The first aircraft is moving S at 250km/h while the second is moving W at 600 km/h-----At what rate is the distance between them...

Two aircraft approach airfield at the same constant altitude.  The first aircraft is moving S at 250km/h while the second is moving W at 600 km/h-----

At what rate is the distance between them changing when the first aircraft is 60 km from the field and the second is 25km from the field.

Asked on by mathmomma

beckden | High School Teacher | (Level 1) Educator

Posted on

The two aircraft form a right triangle with the right ange at the airport.

Suppose the westbound aircraft distance from the airport is x and the southbound aircraft distance is y.  Then we can find the distance between the aircraft using the Pythagorean Theorem.

`s^2 = x^2 + y^2`

Now we need to implicitly differentiate with respect to time to get

` 2s(ds)/(dt) = 2x(dx)/(dt) + 2y(dy)/(dt)`

.

Now `(dx)/(dt) = -600 "km/hr"` and `(dy)/(dt) = -250 "km/hr"`  if we consider this on a coordinate plane.

So `(ds)/(dt) = (x(dx)/(dt) + y(dy)/(dt))/s`

The first aircraft (y) is 60km and (x) is 25km.

`(ds)/(dt) = (25(-600)+(60)(-250))/(sqrt(60^2+25^2)) ~~ -462 "km/hr"`

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