An airplane flying at an altitude of 5 miles passes directly over a radar antenna. When the airplane is 50 miles away (s = 50), the radar detects that the distance s is changing at a rate of 280 miles per hour. What is the speed of the airplane? Round your answer to the nearest integer.
A)281 mi/hr
B)271 mi/hr
C)563 mi/hr
D)141 mi/hr
E)135 mi/hr

Question

Quizplus · Accepted Answer

Using the Pythagorean theorem, we have $s^2 = x^2 + 5^2$, where $s$ is the slant distance from the plane to the radar, and $x$ is the horizontal distance from the radar to the point on the ground directly below the plane. Differentiating both sides with respect to time $t$, we get $2s \frac{ds}{dt} = 2x \frac{dx}{dt}$. Given that $s = 50$ miles and $\frac{ds}{dt} = 280$ mi/hr when the plane is directly above the radar, we can solve for $\frac{dx}{dt}$, which represents the speed of the airplane. Substituting the given values, we find that the speed of the airplane is approximately 281 mi/hr.

An Airplane Flying at an Altitude of 5 Miles Passes