1. ## Airplanes

An airplane is flying at an altitude of 6 miles and passes directly over a radar antenna. When the plane is 10 miles away, the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

The teacher I have loves planes and assigned this problem and many others. I sloved most of them. Thanks for the time

2. by the pythagorean theorem we know that

$6^2+x^2=s^2$ and we want to find $\frac{dx}{dt}$ when x=10.

using the above we know that $s=\sqrt{136}$ when x=10.

taking the derivative with respect to time

$2x\frac{dx}{dt}=2s\frac{ds}{dt} \iff \frac{dx}{dt}=\frac{s}{x}\frac{ds}{dt}=\frac{\sqrt {136}}{10}\frac{240 \mbox{ miles}}{\mbox{ hr}}=48\sqrt{34}\frac{miles}{hr}$