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Question
If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area ?
Solution
Let x be the radius and y be the area of the circular plane.
\[\text { We have } \frac{\bigtriangleup x}{x} = \alpha \text { and } y = x^2 . \]
\[ \Rightarrow \frac{dy}{dx} = 2x\]
\[ \Rightarrow \frac{\bigtriangleup y}{y} = \frac{2x}{y}dx = \frac{2x}{x^2} \times \alpha x = 2\alpha\]
\[\text { Hence, the relative error in the area of the circular plane is } 2\alpha .\]
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