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Solution for The Height of a Cylinder is Equal to the Radius. If an Error of α % is Made in the Height, Then Percentage Error in Its Volume is (A) α % (B) 2α % (C) 3α % (D) None of These - CBSE (Commerce) Class 12 - Mathematics

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Question

The height of a cylinder is equal to the radius. If an error of α % is made in the height, then percentage error in its volume is
(a) α %
(b) 2α %
(c) 3α %
(d) none of these

Solution

(c) 3 \[\alpha\] 

Let x be the radius, which is equal to the height of the cylinder. Let y be its volume.

\[\frac{∆ x}{x} \times 100 = \alpha\]

\[\text { Also }, y = \pi x^2 x = \pi x^3 \left[ \text{ Radius = Height of the cylinder }\right]\]

\[ \Rightarrow \frac{dy}{dx} = 3\pi x^2 \]

\[ \Rightarrow \frac{∆ y}{y} = \frac{3\pi x^2}{y}dx = \frac{3}{x} \times \frac{\alpha x}{100}\]

\[ \Rightarrow \frac{∆ y}{y} \times 100 = 3\alpha\]

\[\text { Hence, the error in the volume of the cylinder is } 3\alpha % .\]

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Solution The Height of a Cylinder is Equal to the Radius. If an Error of α % is Made in the Height, Then Percentage Error in Its Volume is (A) α % (B) 2α % (C) 3α % (D) None of These Concept: Approximations.
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