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प्रश्न
Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube ?
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उत्तर
Let x be the edge of the cube and y be the surface area.
\[y = x^2 \]
\[\text { Let } ∆ x \text { be the error in x and } ∆ y \text { be the corresponding error in } y . \]
\[\text { We have }\]
\[\frac{∆ x}{x} \times 100 = 1\]
\[ \Rightarrow 2x = \frac{x}{100} \left[ \text { Let } dx = ∆ x \right]\]
\[\text { Now }, y = x^2 \]
\[ \Rightarrow \frac{dy}{dx} = 2x\]
\[ \therefore ∆ y = \frac{dy}{dx} \times ∆ x = 2x \times \frac{x}{100}\]
\[ \Rightarrow ∆ y = 2\frac{x^2}{100}\]
\[ \Rightarrow ∆ y = 2\frac{y}{100}\]
\[ \Rightarrow \frac{∆ y}{y} = \frac{2}{100}\]
\[ \Rightarrow \frac{∆ y}{y} \times 100 = 2\]
Hence, the percentage error in calculating the surface area is 2.
