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Solution for Show that the Relative Error in Computing the Volume of a Sphere, Due to an Error in Measuring the Radius, is Approximately Equal to Three Times the Relative Error in the Radius ? - CBSE (Commerce) Class 12 - Mathematics

Question

Show that the relative error in computing the volume of a sphere, due to an error in measuring the radius, is approximately equal to three times the relative error in the radius ?

Solution

Let x be the radius of the sphere and y be its volume.

$\text { Let } ∆ x \text { be the error in the radius and ∆ V be the approximate error in the volume } .$

$y = \frac{4}{3}\pi x^3$

$\Rightarrow \frac{dy}{dx} = 4\pi x^2$

$\Rightarrow ∆ y = dy = \frac{dy}{dx}dx = 4\pi x^2 \times ∆ x$

$\Rightarrow ∆ y = 3 \times \frac{4}{3}\pi x^3 \times \frac{∆ x}{x}$

$\Rightarrow ∆ y = 3 \times y \times \frac{∆ x}{x}$

$\Rightarrow \frac{∆ y}{y} = 3\frac{∆ x}{x}$

Hence proved.

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Solution Show that the Relative Error in Computing the Volume of a Sphere, Due to an Error in Measuring the Radius, is Approximately Equal to Three Times the Relative Error in the Radius ? Concept: Approximations.
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