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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.
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Solution
\[\text { In the right triangle ABC },\]
\[\text { Here,} \]
\[A B^2 + B C^2 = A C^2 \]
\[ \Rightarrow x^2 + \left( 120 \right)^2 = y^2 \]
\[ \Rightarrow 2x\frac{dx}{dt} = 2y\frac{dy}{dt}\]
\[ \Rightarrow \frac{dy}{dt} = \frac{x}{y}\frac{dx}{dt}\]
\[ \Rightarrow \frac{dy}{dt} = \frac{50}{130} \times 52 \left[ \because x = \sqrt{\left( 130 \right)^2 - \left( 120 \right)^2} = 50 \right]\]
\[ \Rightarrow \frac{dy}{dt} = 20 m/\sec\]
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