Question

A kite is moving horizontally at a height of 151.5 m. If the speed of the kite is 10 m/s, how fast is the string being let out, when the kite is 250 m away from the boy flying the kite, if the height of boy is 1.5 m?

Answer 1

We have, height (h) = 151.5 m, speed of kite (v) = 10 m/s
let CD be the height of kite and AB be the height of boy.
Let DB = x m = EA and AC = 250 m
$\therefore \frac{dx}{dt}=10m/s$
from the figure, we see that

EC = 151.5 - 1.5 = 150 m and AE = x
Also, AC = 250 m
in right angled $△CEA,A{E}^2+E{C}^2=A{C}^2\Rightarrow x^2+(150{)}^2=y^2....[1]\Rightarrow x^2+(150{)}^2=(250{)}^2\Rightarrow x^2=(250{)}^2-(150{)}^2=(250+150\left)\right(250-150)=400\times 100\therefore x=20\times 10=200$
From Eq. [1] on differentiating w.r.t. t, we get
$2x.\frac{dx}{dt}+0=2y\frac{dy}{dt}\Rightarrow 2y\frac{dy}{dt}=2x\frac{dx}{dt}\therefore \frac{dy}{dt}=\frac{x}{y}.\frac{dx}{dt}=\frac{200}{250}.10=8$
$therefore$ String is being let out at a rate of 8 m/s.