Hello, asnxbbyx113!
A plane flying horizontally at an altitude of 4 miles and a speed of 440 mph
passes directly over a radar station.
Find the rate at which the distance from the plane to the station is increasing
when it is 10 miles away from the station. Code:
x
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$\displaystyle x$ is the distance the plane has flown horizontally.
$\displaystyle R$ is the plane's distance from the station.
We have: .$\displaystyle R^2 \:=\:x^2 + 4^2$
Differentiate with respect to time: .$\displaystyle 2R\left(\frac{dR}{dt}\right) \:=\:2x\left(\frac{dx}{dt}\right)$
. . and we have: .$\displaystyle \frac{dR}{dt} \:=\:\frac{x}{R}\left(\frac{dx}{dt}\right)$
When $\displaystyle R = 10$, we have: .$\displaystyle 10^2\:=\:x^2 + 16\quad\Rightarrow\quad x \:=\:\sqrt{84}$
. . and we are given: .$\displaystyle \frac{dx}{dt} \,=\,440$
Therefore: .$\displaystyle \frac{dR}{dt}\:=\:\frac{\sqrt{84}}{10}(440) \:=\:88\sqrt{21}$ mph.