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The area of triangle is calculated by the formula $\frac{1}{2} bc\text{ }sin A$ . If the angle A is measure $30^\circ$ with 1% error. Find the % error in area.

(A) $\frac{\pi}{\sqrt{3}}$

(B) $\frac{\pi}{2\sqrt{3}}$

(C) $\frac{2\pi}{\sqrt{3}}$

(D) $\frac{\pi}{2}$

in Calculus | 33 views

Let S denote area,

S=$\frac{1}{2}bcsinA$

Given $\frac{\Delta A}{A}=0.01$

A=$\frac{\pi}{6} \implies \Delta A=0.01 \times \frac{\pi}{6}$

$dS=\frac{1}{2}d(bcsinA)=\frac{1}{2}(csinAdb+bsinAdc+bccosAdA)$

$dS=\frac{1}{2}d(bcsinA)=\frac{1}{2}(cdb+bdc+bccotAdA)$

$dS=\frac{1}{2}d(bcsinA)=\frac{1}{2}(\frac{db}{b}+\frac{dc}{c}+cotAdA)$

Now we have :

$\frac{dS}{S}=\frac{db}{b}+\frac{dc}{c}+\frac{dA}{tanA}$

$\implies \frac{ds}{S}=\frac{0.01 \times \frac{\pi}{6}}{tan30}=0.01 \times \sqrt 3 \times \frac{\pi}{6}$. To match an option maybe there is some approximation.
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