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If the ratio of base radius and height of a cone is 1 : 2 and percentage error in radius $λ %$ , then the error in its volume, is

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a

λ %

b

2 λ %

c

3 λ %

d

none of these

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detailed solution

Correct option is C

let $r$ be the base radius and $h$ be the height of the

cone. Then, $2 r = h .$ Let $V$ be the volume of the cone. Then,

$\begin{array}{l} V = \frac{1}{3} π r^{2} h = \frac{4 π r^{3}}{3} \\ \Rightarrow \frac{d V}{d r} = 4 π r^{2} \\ ∴ Δ V = \frac{d V}{d r} Δ r \\ \Rightarrow Δ V = 4 π r^{2} Δ r \\ \Rightarrow \frac{Δ V}{V} \times 100 = \frac{4 π r^{2}}{4} π r^{3} \times 100 = 3 (\frac{Δ r}{r} \times 100) = 3 λ \end{array}$

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