Question

The length, breadth, and thickness of a strip are $\left(10.0\pm 0.1\right)cm$, $\left(1.00\pm 0.01\right)cm$ and $\left(0.100\pm 0.001\right)cm$ respectively. The most probable error in its volume will be?
(A) $\pm 0.03c{m}^3$
(B) $\pm 0.111c{m}^3$
(C) $\pm 0.012c{m}^3$
(D) None of these.

Answer 1

Hint: In the question, the length, breadth, and thickness are given. So, the volume is given by the product of the length, breadth, and thickness. So the percent error can be found out by the smallest change in length, breadth, and thickness by the formula,
${\displaystyle \frac{\mathrm{\Delta }V}{V}}={\displaystyle \frac{\mathrm{\Delta }l}{l}}+{\displaystyle \frac{\mathrm{\Delta }b}{b}}+{\displaystyle \frac{\mathrm{\Delta }t}{t}}$
Formula used: In this question, we will be using the following formula,
$V=l\times b\times t$
where $V$ is the volume
and $l,b$and $t$ are the length, breadth, and thickness respectively
${\displaystyle \frac{\mathrm{\Delta }V}{V}}={\displaystyle \frac{\mathrm{\Delta }l}{l}}+{\displaystyle \frac{\mathrm{\Delta }b}{b}}+{\displaystyle \frac{\mathrm{\Delta }t}{t}}$
where $\mathrm{\Delta }V,\mathrm{\Delta }l,\mathrm{\Delta }b$ and $\mathrm{\Delta }t$ are the smallest change in volume, length, breadth, and thickness respectively.

Complete step by step answer:
In the question, we are given the length, breadth, and thickness. So we can calculate the volume from the formula, $V=l\times b\times t$.
Now, for the calculation of the most probable error, we need to calculate the errors due to the smallest change in length, breadth, and thickness.
To calculate the most probable error, we use the formula
${\displaystyle \frac{\mathrm{\Delta }V}{V}}={\displaystyle \frac{\mathrm{\Delta }l}{l}}+{\displaystyle \frac{\mathrm{\Delta }b}{b}}+{\displaystyle \frac{\mathrm{\Delta }t}{t}}$
From the question, we are given the values as,
$l=10cm$ and $\mathrm{\Delta }l=0.1cm$
$b=1cm$ and $\mathrm{\Delta }b=0.01cm$
$t=0.1cm$ and $\mathrm{\Delta }t=0.001cm$
So substituting these values in the equation we get,
${\displaystyle \frac{\mathrm{\Delta }V}{V}}={\displaystyle \frac{0.1}{10}}+{\displaystyle \frac{0.01}{1}}+{\displaystyle \frac{0.001}{0.1}}$
Now, the values of
${\displaystyle \frac{0.1}{10}}={\displaystyle \frac{0.01}{1}}={\displaystyle \frac{0.001}{0.1}}=0.01$
So from the equation, we have
${\displaystyle \frac{\mathrm{\Delta }V}{V}}=0.01+0.01+0.01$
On adding the values we get,
${\displaystyle \frac{\mathrm{\Delta }V}{V}}=0.03$
Therefore the most probable error in volume will be $\pm 0.03c{m}^3$.
So the correct answer is option (A); $\pm 0.03c{m}^3$.

Note:
In any formula, the most probable error is given by the sum of the errors in all the variables. These errors may be caused due to a lot of factors like errors in measurement of the individual variables due to problems in an instrument or taking a reading incorrectly, physical conditions like temperature, and various other factors.