1. ## AIRPLANE Problem

Two airplanes (at the same height) are flying away from an airport at a right angle to each other.
The distance between them is 470 miles.
Plane X is 270 miles from the airport.
Plane Y is traveling at a speed of 520 mi/hr.
The distance between them is increasing by 660 mi/hr.

How fast is plane X traveling?

2. Hello, McDiesel!

Two airplanes are flying away from an airport at a right angle to each other.
The distance between them is 470 miles.
Plane $\displaystyle X$ is 270 miles from the airport.
Plane $\displaystyle Y$ is traveling at a speed of 520 mi/hr.
The distance between them is increasing by 660 mi/hr.

How fast is plane $\displaystyle X$ traveling?
Code:
    Q *
: |  *
y |     *
: |        *
C *           *   z
: |  *           *
___|     *  470      *
20√370|        *           *
: |           *           *
: |              *           *
- * - - - - - - - - * - - - - - *
A       270       B     x     P

The airport is at $\displaystyle A.$

Plane $\displaystyle X$ is at $\displaystyle B\!:\;AB = 270$

Plane $\displaystyle Y$ is at $\displaystyle C\!:\;BC = 470.$
In right triangle $\displaystyle CAB\!:\;\;AC^2 + 270^2 \:=\:470^2 \quad\Rightarrow\quad AC \:=\:20\sqrt{370}$

In a certain time, plane $\displaystyle X$ flew $\displaystyle x$ miles from $\displaystyle B$ to $\displaystyle P.$
. . $\displaystyle AP \:=\:x+270$

In a certain time, plane $\displaystyle Y$ flew $\displaystyle y$ miles from $\displaystyle C$ to $\displaystyle Q\!:\;\;\tfrac{dy}{dt} = 520\text{ mph}$
. . $\displaystyle AQ \:=\:y + 20\sqrt{370}$

The distance between them is: .$\displaystyle z \:=\:PQ\:\text{ and }\:\tfrac{dz}{dt} = 660\text{ mph}$
From right triangle $\displaystyle QAP\!:\;z^2 \;=\;(x+270)^2 + (y + 20\sqrt{370})^2$

Differentiate with respect to time: .$\displaystyle 2z\,\frac{dz}{dt} \;=\;2(x+270)\,\frac{dx}{dt} + 2(y + 20\sqrt{370})\,\frac{dy}{dt}$

. . and we have: .$\displaystyle (x+270)\,\frac{dx}{dt} \;=\;z\,\frac{dz}{dt} - (y + 20\sqrt{370})\,\frac{dy}{dt}$

At that instant in question: . $\displaystyle \begin{Bmatrix}x &=& 0 \\ y&=&0 \\ z &=& 470 \\ \frac{dy}{dt} &=& 520 \\ \\[-4mm] \frac{dz}{dt} &=& 660 \end{Bmatrix}$

Substitute those values and solve for $\displaystyle \frac{dx}{dt}$