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प्रश्न
If the radius of a sphere is measured as 9 cm with an error of 0.03 m, find the approximate error in calculating its surface area ?
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उत्तर
Let x be the radius and y be the surface area of the sphere.
\[\text { Then }, \]
\[x = 9\]
\[ ∆ x = 0 . 03 m = 3cm\]
\[ \Rightarrow x + ∆ x = 9 + 3 = 12 cm\]
\[y = 4 \pi x^2 \]
\[\text { For } x = 9, \]
\[ y = 4\pi \times 9^2 = 324\pi\]
\[\frac{dy}{dx} = 8\pi x\]
\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 9} = 72\pi\]
\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = 72\pi \times 3 = 216\pi {cm}^2 \]
\[\text { Therefore, the approximate error in the surface area is} 216\pi c m^2 . \]
\[\text { Disclaimer: This solution has been created according to the question given in the book . However, the solution given in the book is incorrect } .\]
