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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 by (485 points) | 33 views

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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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