Question

The density of a cube is found by measuring its mass and the length of its side. If the maximum errors in the measurement of mass and length are 0.3% and 0.2% respectively, the maximum error in the measurement of density is:
A. 0.3%
B. 0.5%
C. 0.9%
D. 1.1%

Answer 1

Hint: Our very first task is to find the expression for density in terms mass and length. From this expression we could easily deduce the expression for finding the maximum error in measurement. You could then substitute the percentage error in measurements of mass and length in the above expression and hence find the answer.

Formula used: Density,
$\rho =\dfrac{M}{V}=\dfrac{M}{{{L}^{3}}}$

Complete step by step answer:
In the question, we are given the case of measurement of the density of a cube. This measurement is made by taking the measurements of the cube’s mass and length of its edge. The maximum errors in these measurements are found to be 0.3% and 0.2% respectively. We are supposed to find the maximum error in the measurement of density.
Let us recall the expression for density in terms of mass and length. Density is defined as the ratio mass to volume. Mathematically,
$\rho =\dfrac{M}{V}=\dfrac{M}{{{L}^{3}}}$
The error in the measurement of density could be found by using,
$\dfrac{\Delta \rho }{\rho }=\left( \dfrac{\Delta M}{M} \right)+3\left( \dfrac{\Delta L}{L} \right)$
But,
$\dfrac{\Delta M}{M}=0.3%$
$\dfrac{\Delta L}{L}=0.2%$
Substituting we get,
$\dfrac{\Delta \rho }{\rho }=\left( 0.3 \right)+3\left( 0.2 \right)$
$\therefore \dfrac{\Delta \rho }{\rho }=0.9%$

So, the correct answer is “Option C”.

Note: What we should always remember is that while calculating the percentage error we should take the sum of the individual errors. But however here one of the terms is raised to some power. Thus, we should multiply that power with the error of the quantity’s measurement while taking the sum.