# Find the rate at which the distance between the plane and the station is increasing when the plane is 3 km away from the station. A plane flying horizontally at an altitude of 2 km and a speed of...

Find the rate at which the distance between the plane and the station is increasing when the plane is 3 km away from the station.

A plane flying horizontally at an altitude of 2 km and a speed of 800 km/h passes directly over a radar station. (We are looking at the instant when the direct “air” distance between the plane and the station is 3 km.)

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The plane flying horizontally at an altitude of 2 km at a speed of 800 km/h passes over a radar station. After a time t, the distance between the plane and the radar station is `D = sqrt(2^2 + (800*t)^2)` = `sqrt(4 + 640000*t^2)`

The rate at which D is changing with time is `(dD)/(dt) = (1/2)*1280000*t*(1/sqrt(4 + 640000*t^2))`

=> `640000*t/sqrt(4 + 640000*t^2)`

If D = 3, `sqrt(4 + 640000*t^2) = 3`

=> `4 + 640000*t^2 = 9`

=> `640000*t^2 = 5`

=> `t = sqrt 5/800`

At `t = sqrt 5/800` , `(dD)/(dt) = (640000*(sqrt 5/800))/(sqrt(4 + (640000*5/640000)))`

=> `(dD)/(dt) = (800*(sqrt 5))/(sqrt9)`

=> `800*sqrt 5/3`

**The rate at which the distance of the plane from the radar station is increasing is `(800*sqrt5)/3` **