Question

The density of a cube is measured by measuring its mass and length of 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

Hint: We first need to know about the relative error and the maximum error. Then we can easily solve this type of problem. Relative error is the ratio of the magnitude of error to the actual value of the physical quantity. Like relative error in mass is given by $\dfrac{\triangle m}{m}$ where $\triangle m$ is the magnitude of the error that is
$\triangle m=|m(actual)-m(measured)|$.
Now, maximum error means that whenever two relative errors are together, they will be added but not subtracted.

Formula used: $\rho=\dfrac{m}{a^3}$
$\dfrac{\triangle\rho}{\rho}=\dfrac{\triangle m}{m}+3\dfrac{\triangle a}{a}$

Complete step-by-step answer:
Mass of the cube=m, length of the side of the cube=a. Volume of the cube, $V=a^3$, Density of the cube= $\rho$
$\rho=\dfrac{m}{V}=\dfrac{m}{a^3}$
Taking logarithms,
$ln\rho=ln m-3ln a$
Differentiating this,
$\dfrac{\triangle\rho}{\rho}=\dfrac{\triangle m}{m}-3\dfrac{\triangle a}{a}$
Now since error has to be maximum,
$\dfrac{\triangle\rho}{\rho}=\dfrac{\triangle m}{m}+3\dfrac{\triangle a}{a}$
Now, the given data are
$\dfrac{\triangle m}{m}=4$ and $\dfrac{\triangle a}{a}=3$
Putting this values in the above formula, $\dfrac{\triangle\rho}{\rho}=0.04+3×0.03=0.13=13\%$
Hence option D is the correct answer.

Note: The most common mistake that is made in this type of questions, is that often students subtract the two relative errors, i.e. the errors in mass and in length of the side of the cube. This is to be kept in mind that errors can only be added.
Also remember that, the magnitude of the error is always positive.