Question

A kite is $120m$ high and $130m$ of string is out. If the kite is moving away horizontally at the rate of $52m/s$, find the rate at which the string is being paid out.

Answer 1

Let at any time $t$ A be the position of kite and AC is position of string. Now, from figure,

$A{C}^2=A{B}^2+B{C}^2$

$y^2+x^2+{120}^2...........\left(1\right)$

Now, lite is moving away horizontally at the rate of 52 m/s.

So,

$\frac{dx}{dt}=52m/s$

Differentiating equation (1) wrt t ,we get

$2y\frac{dy}{dt}=2x\frac{dx}{dt}+0$

$\frac{dy}{dt}=\frac{x}{y}\left(\frac{dx}{dt}\right)$

$\frac{dy}{dt}=\frac{x}{y}\left(52\right)$

When $y=130m$ and $x=50m$

$\frac{dy}{dt}=\frac{50}{130}\left(52\right)=20m/sec$

Hence, string is paid out at the rate of 20 m/sec