If the relative error in measuring the radius of a circular plane is α,
Question:

If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area.

Solution:

Let x be the radius and y be the area of the circular plane.

We have $\frac{\Delta x}{x}=\alpha$ and $y=x^{2}$

$\Rightarrow \frac{d y}{d x}=2 x$

$\Rightarrow \frac{\triangle y}{y}=\frac{2 x}{y} d x=\frac{2 x}{x^{2}} \times \alpha x=2 \alpha$

Hence, the relative error in the area of the circular plane is $2 \alpha$.

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