# Calculus related rates

• Jan 6th 2007, 04:33 PM
asnxbbyx113
Calculus related rates
A plane flying horizontally at a altitude of 4mi and a speed of 440mi/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 10mi away from the station.

Thanks!
• Jan 6th 2007, 09:21 PM
Soroban
Hello, asnxbbyx113!

Quote:

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       + - - - - - - - - - - - *       |                    *       |                *     4 |              *       |          * R       |        *       |    *       |  *       * - - - - - - - - - - - -

$\displaystyle x$ 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.