A plane flying horizontally at an altitude of 1 mi and a speed of 540 mi/h passes directly over a radar station. How do you find the rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station?

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Answer 1 · 2016-12-23T20:33:48

Rate $~~ 529.1$ mi/hour

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Let us set up the following variables:

${(s, "Horizontal distance of plane from the radar station (mi)"), (x, "Actual direct distance of plane from the radar station (mi)") :}$

Then, our aim is to find $dx/dt$ when $x=5$

The plane is moving in the horizontal direction at constant speed ( $540$ mi/hour). i.e $(ds)/dt=540$

By Pythagoras;

$\ \ \ \ \ x^2 = s^2+1^2$
$:.x^2 = s^2+1$ ..... [1]

Differentiating Implicitly wrt $t$ we get:

$2xdx/dt = 2s(ds)/dt + 0$
$\ \ xdx/dt = s(ds)/dt$
$\ \ xdx/dt = 540s$
$\ \ xdx/dt = 540sqrt(x^2-1) " "$ (Using [1])

When $x=5 =>$

$\ \ \ \ \ 5dx/dt = 540sqrt(25-1)$
$:. 5dx/dt = 540sqrt(24)$
$:. dx/dt = 108sqrt(24)$
$:. dx/dt ~~ 529.1$ mi/hour