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Question
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is
Options
\[8\sqrt{3} \ {cm}^2 /hr\]
\[4\sqrt{3} \ {cm}^2 /hr\]
\[\frac{\sqrt{3}}{8} \ {cm}^2 /hr\]
none of these
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Solution
\[8\sqrt{3} \ {cm}^2 /hr\]
\[\text { Let x be the side and A be the area of the equilateral triangle at any time t. Then },\]
\[A = \frac{\sqrt{3}}{4} x^2 \]
\[ \Rightarrow \frac{dA}{dt} = \frac{\sqrt{3}}{2}x\left( \frac{dx}{dt} \right)\]
\[ \Rightarrow \frac{dA}{dt} = \frac{\sqrt{3}}{2}\left( 2 \right)\left( 8 \right)\]
\[ \Rightarrow \frac{dA}{dt} = 8\sqrt{3} \ {cm}^2 /hr\]
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