Question

A plane flying horizontally at an altitude of 3 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station. (Round your answer to the nearest whole number.)

Answer 1

`(dr)/(dt) = 400\,(mi)/(h)`

Explanation:

The distance from the plane to the station is given by the Pyhtagorean Theorem:

`r^(2) = x^(2)+y^(2)`

Where x and y are the horizontal and vertical distances, respectively.

The rate of change of the distance is obtained by implicit differentiation:

`2\cdot r \cdot (dr)/(dt) = 2\cdot x \cdot (dx)/(dt) + 2\cdot y \cdot (dy)/(dt)`

`(dr)/(dt) = (x\cdot (dx)/(dt) + y\cdot (dy)/(dt) )/(r)`

`(dr)/(dt) = \frac{x\cdot (dx)/(dt)+y\cdot (dy)/(dt) }{\sqrt{x^(2)+y^(2)} }`

`(dr)/(dt) = \frac{(4\,mi)\cdot \left(500\,(mi)/(h) \right)+(3\,mi)\cdot \left(0\,(mi)/(h) \right)}{\sqrt{(3\,mi)^(2)+(4\,mi)^(2)} }`

`(dr)/(dt) = 400\,(mi)/(h)`