Question

If there is an error of $\mathrm{k}\%$ in measuring the edge of a cube, then the percent error in estimating its volume is

• A. $\mathrm{k}$
• B. $3\mathrm{k}$
• C. $\frac{\mathrm{k}}{3}$
• D. None of these

Answer 1

The correct option is B

$3\mathrm{k}$

Explanation for Correct answer:

Option (B) correct answer

Step 1 Given data:

The error of $\mathrm{k}\%$ in measuring the edge of a cube.

Step 2 Formula used:

The volume of a cube is ${\mathrm{x}}^3$, where $\mathrm{x}$ is the edge length of the cube.

Step 3 Percent error:

The error of $\mathrm{k}\%$ in measuring the edge of a cube, that is $\frac{\mathrm{dx}}{\mathrm{x}}\times 100=\mathrm{k}....\left(1\right)$

The volume of a cube is $\mathrm{v}={\mathrm{x}}^3....\left(2\right)$

Therefore,

$$\begin{gathered} \frac{\mathrm{dv}}{\mathrm{dx}}=\frac{\mathrm{d}\left({\mathrm{x}}^3\right)}{\mathrm{dx}} \\ \frac{\mathrm{dv}}{\mathrm{dx}}=3{\mathrm{x}}^2 \\ \mathrm{dv}=3{\mathrm{x}}^2\mathrm{dx} \end{gathered}$$

The percent error in estimating its volume is,

$$\begin{gathered} \frac{\mathrm{dv}}{\mathrm{v}}\times 100=\frac{3{\mathrm{x}}^2\mathrm{dx}}{{\mathrm{x}}^3}\times 100 \\ \frac{\mathrm{dv}}{\mathrm{v}}\times 100=\frac{3\mathrm{dx}}{\mathrm{x}}\times 100 \\ \frac{\mathrm{dv}}{\mathrm{v}}\times 100=3\left(\frac{\mathrm{dx}}{\mathrm{x}}\right)\times 100....\left(3\right) \end{gathered}$$

Substituting equation (1) in equation (3),

$$\begin{gathered} \frac{\mathrm{dv}}{\mathrm{v}}\times 100=3\left(\frac{\mathrm{dx}}{\mathrm{x}}\right)\times 100 \\ \frac{\mathrm{dv}}{\mathrm{v}}\times 100=3\mathrm{k}\% \end{gathered}$$

Therefore, the percent error in estimating the volume of a cube is $3\mathrm{k}\%$.