An airplane flying at an altitude of 6 miles passes directly over a radar antenna. When the airplane is 10 miles away (s=10) the radar detects the distance s is changing at a rate of 240 miles per hour. What is the speed of the airplane?
An airplane flying at an altitude of 6 miles passes directly over a radar antenna. When the airplane is 10 miles away (s=10) the radar detects the distance s is changing at a rate of 240 miles per hour. What is the speed of the airplane?
Let the horizontal distance of the plane be x at time t.
$\displaystyle s^2 = x^2 + 6^2$ .... (1)
$\displaystyle \Rightarrow 2s \frac{ds}{dt} = 2x \frac{dx}{dt}$
$\displaystyle \Rightarrow \frac{dx}{dt} = \frac{s}{x} \cdot \frac{ds}{dt}$ .... (2)
Use equation (1) to solve for $\displaystyle x$ when $\displaystyle s = 10$. You're given $\displaystyle \frac{ds}{dt} = 240$ miles per hour when $\displaystyle s = 10$ miles. Substitute all this into equation (2).