A plane flying horizontally at an altitude of $1 \mathrm { mi }$ and a speed of $550 \mathrm { mi } / \mathrm { h }$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is $2 \mathrm { mi }$ away from the station.
A) $\approx 476 \mathrm { mi } / \mathrm { h }$
B) $\approx 670 \mathrm { mi } / \mathrm { h }$
C) $\approx 455 \mathrm { mi } / \mathrm { h }$
D) $\approx 570 \mathrm { mi } / \mathrm { h }$
E) $\approx 495 \mathrm { mi } / \mathrm { h }$

Question

Quizplus · Accepted Answer

Using the Pythagorean theorem, the distance $s$ from the plane to the station is given by $s^2 = x^2 + 1^2$, where $x$ is the horizontal distance from the station. Differentiating with respect to time $t$, we get $2s \frac{ds}{dt} = 2x \frac{dx}{dt}$. Given $x = \sqrt{3}$ when $s = 2$ and $\frac{dx}{dt} = 550$, solve for $\frac{ds}{dt}$ to find it is approximately $476 \mathrm{mi/h}$.

A Plane Flying Horizontally at an Altitude Of 1mi1 \mathrm { mi }1mi

A Plane Flying Horizontally at an Altitude Of $1 \mathrm { mi }$