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The side of an equilateral triangle is 5 units. In measuring the side, an error of 0.05 units is made. Then the percentage error in measuring the area of the triangle is
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Verified Answer
The correct answer is:
2
Let $A$ be the area and $x$ be the side of an equilateral triangle.
$$
\therefore \quad A=\frac{\sqrt{3}}{4} x^2
$$
On differentiation $\frac{d A}{d x}=\frac{\sqrt{3}}{4}(2 x)$
$$
\Rightarrow \quad d A=\frac{\sqrt{3}}{2} x \cdot d x \Rightarrow \Delta A=\frac{\sqrt{3}}{2} \cdot x \cdot \Delta x
$$
$\therefore$ Percentage error in area $=\Delta A / A \times 100$
$$
\begin{aligned}
& =\frac{\sqrt{3} / 2 \times \Delta x \times x \times 100}{\sqrt{3} / 4 \times x^2}=\frac{2 \Delta x}{x} \times 100 \\
& =\frac{2 \times 0.05}{5} \times 100=\frac{2 \times 5}{5 \times 100} \times 100=2
\end{aligned}
$$
$$
\therefore \quad A=\frac{\sqrt{3}}{4} x^2
$$
On differentiation $\frac{d A}{d x}=\frac{\sqrt{3}}{4}(2 x)$
$$
\Rightarrow \quad d A=\frac{\sqrt{3}}{2} x \cdot d x \Rightarrow \Delta A=\frac{\sqrt{3}}{2} \cdot x \cdot \Delta x
$$
$\therefore$ Percentage error in area $=\Delta A / A \times 100$
$$
\begin{aligned}
& =\frac{\sqrt{3} / 2 \times \Delta x \times x \times 100}{\sqrt{3} / 4 \times x^2}=\frac{2 \Delta x}{x} \times 100 \\
& =\frac{2 \times 0.05}{5} \times 100=\frac{2 \times 5}{5 \times 100} \times 100=2
\end{aligned}
$$
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