Question

In the density measurement of a cube, the mass and edge length are measured as (10.00 $\pm$ 0.10) kg and (0.10 $\pm$ 0.01) m, respectively. The error in the measurement of density:
A. $0.10\dfrac{{kg}}{{{m^3}}}$
B. $0.31\dfrac{{kg}}{{{m^3}}}$
C. $0.07\dfrac{{kg}}{{{m^3}}}$
D. $0.01\dfrac{{kg}}{{{m^3}}}$
E. No answer

Answer 1

Hint: The relation of density with mass and volume is basic and very important to solve this question. The given values contain the actual value with the error in that measurement. Error formula is helpful here.

Complete step-by-step answer:
We know that density is the ratio of mass and volume of the system. Here the system is cube then the volume of the cube is $A^3$ where A is the edge length of the cube.
Then the formula of density is,
$Density = \dfrac{{mass}}{{volume}}$
$d = \dfrac{m}{{{A^3}}}$
It is given that error in mass and edge length of the cube.
Let A and ∆A be the edge length and error in edge length respectively. Similarly, m and ∆m be the mass and error in mass respectively.
The formula to find out error in the density will be,
$\dfrac{{\Delta d}}{d} = \dfrac{{\Delta m}}{m} + \dfrac{{3\Delta A}}{A}$
Here, A=0.10m , ∆A=0.01m and m=10kg, ∆m=0.10kg
Putting these values in above equation we get,
$\dfrac{{\Delta d}}{d} = \dfrac{{0.10}}{{10}} + \dfrac{{3 \times 0.01}}{{0.10}}$
$\dfrac{{\Delta d}}{d} = 0.31\dfrac{{kg}}{{{m^3}}}$
So, the error in the measurement of the density of the cube is $\Delta d = 0.31\dfrac{{kg}}{{{m^3}}}$
Correct option is B.

Note: Units are very important here. Determine which value is error from given values. If you are done correctly and appropriately apply the error formula then the answer will be correct otherwise some silly mistakes can give the wrong answer.