Question

An aeroplane at an altitude of $960meters$ flying horizontally at $720km/hr$. passes directly over an observer. The rate at which it is approaching the observers when it is $1600meters$ directly away from him is

• A. $576km/hr$
• B. $676km/hr$
• C. $720km/hr$
• D. $570km/hr$

Answer 1

The correct option is D $570km/hr$

Let $A$ be position of aeroplane & $B$ is position of observer at time $t$

$OA=960m$

$OB=x$, horizontal distance &

$AB=y$, actual distance of aeroplane from observer at time $t$

$\frac{dx}{dt}=720km/hr$ is rate at which aeroplane is moving over an observer &

$\frac{dy}{dt}=$ rat6e at which aeroplane is approaching to the observer

$y^2=x^2+(960{)}^2=x^2+(0.960{)}^2-----\left(i\right)$

Differentiating art $t$, we get

$2y\frac{dy}{dt}=\frac{x}{y}.\frac{dx}{dt}$

$\to y=1600m=\frac{16}{10}=\frac{8}{6}km$

$\therefore \frac{64}{25}=x^2+(0.96{)}^2$

$x=\sqrt{\frac{64}{25}-(0.96{)}^2},x=\sqrt{\frac{40}{25}}=\frac{2\sqrt{10}}{50}$

$\therefore \frac{dy}{dt}=\left(\frac{2\sqrt{10}}{5}\right)\left(\frac{5}{8}\right)\times 720$

$=180\sqrt{10}=569.2km/hr$

$\therefore$ Option $D$ is correct