Question

The density of a cube is measured by measuring its mass and the length of the sides. If the maximum error in the measurements of mass and length are $4\%$ and $3\%$ respectively, the maximum error in the measurement of density will be

• A. $7\%$
• B. $9\%$
• C. $12\%$
• D. $13\%$

Answer 1

The correct option is D

$13\%$

Step 1. Given data:

Let, the mass of the cube = $m$

length of each side of the cube is = $a$

Density of the cube = $\rho$

Given, The maximum error in the measurements of mass $\frac{\left(∆m\right)}{m}=0.04$

and maximum error in the measurements of Length. $\frac{\left(∆a\right)}{a}=0.03$

Step 2. Formula used:

Volume of the cube $v=a^3$

Density= mass/volume $\left(\rho \right)$= $\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$v$}\right.$

Step 2. Calculations:

Now, $\rho =\left(\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$v$}\right.\right)=\left(\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$a^3$}\right.\right)$

Since error has to be maximum

So, $\left(\raisebox{1ex}{$∆\rho $}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.\right)=\left(\raisebox{1ex}{$∆m$}\!\left/ \!\raisebox{-1ex}{$m$}\right.\right)+3\left(\raisebox{1ex}{$∆a$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)$

Now putting the given values, we get

$$\begin{gathered} \left(\raisebox{1ex}{$∆\rho $}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.\right)=0.04+3\left(0.03\right)=0.13 \\ \left(\raisebox{1ex}{$∆\rho $}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.\right)\times 100=13\% \end{gathered}$$

Therefore, the correct option is option D.