Question 12 Marks
If the relative error in measuring the radius of a circular plane is $\alpha,$ find the relative error in measuring its area.
Answer
View full question & answer→Let x be the radius and y be the area of the circular plane.
We have $\frac{\triangle\text{x}}{\text{x}}=\alpha\ \text{and}\ \text{y}=\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=2\text{x}$
$\Rightarrow\frac{\triangle\text{y}}{\text{y}}=\frac{2\text{x}}{\text{y}}\text{dx}=\frac{2\text{x}}{\text{x}^2}\times\text{ax}=2\alpha$
Hence, the relative error in the area of the circular plane is $2\alpha$
We have $\frac{\triangle\text{x}}{\text{x}}=\alpha\ \text{and}\ \text{y}=\text{x}^2$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=2\text{x}$
$\Rightarrow\frac{\triangle\text{y}}{\text{y}}=\frac{2\text{x}}{\text{y}}\text{dx}=\frac{2\text{x}}{\text{x}^2}\times\text{ax}=2\alpha$
Hence, the relative error in the area of the circular plane is $2\alpha$