Question

If the error committed in measuring the radius of the circle is $0.05\%$, then the corresponding error in calculating the area is

• A. $0.05\%$
• B. $0.25\%$
• C. $0.1\%$
• D. $0.2\%$

Answer 1

The correct option is C

$0.1\%$

Step 1 Given data:

The error committed in measuring the radius of the circle is $0.05\%$.

Step 2 Formula used:

The area of the circle is $A={\mathrm{\pi r}}^2$, where $\mathrm{r}$ is the radius of the circle.

Step 3 Error in calculating the area:

The error in the radius of the circle is $0.05\%$

That is,

$\frac{\mathrm{dr}}{\mathrm{r}}=0.05...\left(1\right)$

Area is, $A={\mathrm{\pi r}}^2......\left(2\right)$

$$\begin{gathered} \frac{dA}{dr}=\frac{d\left({\mathrm{\pi r}}^2\right)}{dr} \\ \frac{dA}{dr}=2\mathrm{\pi r} \\ \mathrm{dA}=2\mathrm{\pi rdr}....\left(3\right) \end{gathered}$$

Divide equation (3) by equation (2),

$$\begin{gathered} \frac{\mathrm{dA}}{\mathrm{A}}=\frac{2\mathrm{\pi rdr}}{{\mathrm{\pi r}}^2} \\ \frac{\mathrm{dA}}{\mathrm{A}}=\frac{2\mathrm{dr}}{\mathrm{r}}.....\left(4\right) \end{gathered}$$

Substituting equation (1) in equation (4)

That is,

$$\begin{gathered} \frac{\mathrm{dA}}{\mathrm{A}}=2\times \frac{dr}{r} \\ \frac{\mathrm{dA}}{\mathrm{A}}=2\times 0.05 \\ \frac{\mathrm{dA}}{\mathrm{A}}=0.01 \end{gathered}$$

Therefore, the error committed in measuring the radius of the circle is $0.05\%$, then the corresponding error in calculating the area is $0.1\%$. Hence, option C is correct.