Question

What is the absolute uncertainty of c where c is: \[c=\left( 10.0\pm 0.1m \right)\centerdot \left( 120.2\pm 0.2m \right)\]
$1.)0.45{{m}^{2}}$
$2.)14{{m}^{2}}$
$3.)0.5{{m}^{2}}$
$4.)0.3{{m}^{2}}$

Answer 1

Hint: Firstly, find the absolute value of $C$with the help of this equation:-\[C=A\times B\].After that, by multiplying both sides of this formula :-$\dfrac{\Delta C}{C}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$with $C$,we will able to get the desired answer or result.


Formula Used: $\dfrac{\Delta C}{C}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$
Where $'\Delta C','\Delta A'and'\Delta B'$ are the uncertainty values in the absolute value of ‘C’, ‘A’ ,and ‘B’.

Complete step-by-step solution:
Since, we have to find the absolute value of C.
Therefore,\[C=A\times B\]
So, $\Rightarrow C=10\times 120.2$
$\therefore C=1202$
Now, we have to find the value of uncertainty in C. That is$\Delta C$,
Therefore, $\Rightarrow \dfrac{\Delta C}{C}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$
Multiplying both sides with $C$.We get,
$\Rightarrow \dfrac{\Delta C}{C}\times C=\left( \dfrac{\Delta A}{A}+\dfrac{\Delta B}{B} \right)\times C$
\[\Rightarrow \Delta C=\left( \dfrac{\Delta A}{A}+\dfrac{\Delta B}{B} \right)\times C\]
\[\Rightarrow \Delta C=\left( \dfrac{0.1}{10}+\dfrac{0.2}{120.2} \right)\times 1202\]
\[\Rightarrow \Delta C=\left( 0.01+0.0016639 \right)\times 1202\]
\[\Rightarrow \Delta C=\left( 0.0116639 \right)\times 1202\]
$\therefore \Delta C=14.02$
So, correct answer is $\Delta C=14.02$
Hence, the correct option for absolute uncertainty of c is option 2 which is $14{{m}^{2}}$.
Additional information: Uncertainty as used here means the range of possible values within which the true value of the measurement that has been done lies. This definition has changed the usage of some other generally used terms. Let’s take an example, the term accuracy is often used to mean the difference between a measured result and the actual or you can call it as true value. Since, either the true value of a measurement or the accuracy of a measurement is usually not known. Because of these definitions, we modified how to report lab results. For example, when students report the results of lab measurements, they do not calculate a percent error between their result and the actual value. Instead, they determine whether the accepted value falls within the range of given uncertainty of their result or not.

Note: Always first find the value of the absolute value of C. After then only, multiply the resulting value of the absolute value of C with the expression of uncertainty to get the result. Falling to follow the manner as prescribed above will lend you the incorrect result.