Question

The length of cylinder is $10.0\,cm$ and its radius is $4.00\,cm$ The percentage error in volume of cylinder will be:
A. $1.5\%$
B. $2\%$
C. $3.5\%$
D. $10.7\%$

Answer 1

In order to solve this question we need to understand the definition of cylinder which states that it is a closed shape enclosed by two parallel discs and curved rectangle. Now error is defined as the difference between actual value and calculated value. Percentage error of a quantity is the percentage of ratio of the calculated value to the original value. Volume represents the actual amount of space covered by an object.

Formula used:
Percentage error of any physical quantity depends upon other parameter as $Q = {a^x}{b^y}$
$\% Q = x(\% a) + y(\% b)$ which means the percentage error of quantity $Q$ is equal to $x$ times the percentage error in quantity $a$ and add with $y$ times percentage error in quantity $b$.
Volume of cylinder is $V = \pi {r^2}h$

Complete step by step answer:
According to the question, we have given that radius of cylinder is $r = 4.00cm$ since, after decimal point the measurement accuracy is up to two decimals hence the minimum error in radius will be of $\Delta r = 0.01cm$ so,
Percentage error in radius will be written as
$\% r = \dfrac{{\Delta r}}{r} \times 100$
$\Rightarrow \% r = \dfrac{{0.01}}{4} \times 100$
$\Rightarrow \% r = 0.25 \to (i)$

Now, the height of cylinder is given $h = 10.0cm$ here, the measurement accuracy is up to one decimal point, so minimum error in height will be of $\Delta h = 0.1cm$ so,
percentage error in height will be written as
$\% h = \dfrac{{\Delta h}}{h} \times 100$
$\Rightarrow \% h = \dfrac{{0.1}}{{10}} \times 100$
$\Rightarrow \% h = 1 \to (ii)$
Now using formula of percentage error we have,
$V = \pi {r^2}h$ so using formula $\% Q = x(\% a) + y(\% b)$ we get,
$\% V = 2(\% r) + (\% h)$
On putting the value from equation $(i)and(ii)$ we get,
$\% V = 2(0.25) + (1)$
$\therefore \% V = 1.5$
So, the error in volume of cylinder is $1.5\%$

Hence, the correct option is A.

Note: It should be remembered that percent error is totally laboratory based instrumental error we can minimize the error by correcting technical issues but we can’t remove it because of Heisenberg uncertainty principle which states that conjugate pair like position momentum, energy time could not be measured simultaneously because Heisenberg Uncertainty Principle is inherent error in body but this is only viewed when we deal at microscopic level as at microscopic level matter waves do not interact with each other. Also mathematically we can calculate that error and add or subtract from the observed result so to get the correct answer.