Question:
A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out.
Solution:
In the right triangle $A B C$,
Here,
$A B^{2}+B C^{2}=A C^{2}$
$\Rightarrow x^{2}+(120)^{2}=y^{2}$
$\Rightarrow 2 x \frac{d x}{d t}=2 y \frac{d y}{d t}$
$\Rightarrow \frac{d y}{d t}=\frac{x}{y} \frac{d x}{d t}$
$\Rightarrow \frac{d y}{d t}=\frac{50}{130} \times 52$ $\left[\because x=\sqrt{(130)^{2}-(120)^{2}}=50\right]$
$\Rightarrow \frac{d y}{d t}=20 \mathrm{~m} / \mathrm{sec}$