Question

It is given that \[A={{B}^{2}}\]. If \[A=100\pm 0.20\] then B is equal to:
A. \[10\pm 0.20\]
B. \[10\pm 0.02\]
C. \[10\pm 0.01\]
D. \[10\pm 0.1\]

Answer 1

Hint: In this question we are asked to find the value of B from the given equation. To solve this question, we will be calculating the actual value of B from the given equation. Later, using logarithmic functions we will calculate the error value for B. Thus, we will therefore calculate our final answer including both the actual and error value of B.

Complete step-by-step answer:
it is given to us that \[A=100\pm 0.20\]. Here A = 100 and the error value of A is 0.2.
Now, to calculate the value of B, we have been given the expression
\[A={{B}^{2}}\] ……………….. (1)
Substituting the value of A
We get,
\[100={{B}^{2}}\]
Therefore,
\[B=10\]
Now to calculate the error value in B
Let us take the log of equation (1)
We get,
\[\log A=\log {{B}^{2}}\]
Also written as,
\[\log A=2\log B\]
Differentiating both sides
We get,
\[\dfrac{dA}{A}=2\dfrac{dB}{B}\]
It can be written as,
\[\dfrac{\Delta A}{A}=2\dfrac{\Delta B}{B}\]
Here \[\Delta A\] and \[\Delta B\] are the error values in A and B respectively.
Therefore, after substituting the values
We get,
\[\dfrac{0.2}{100}=2\dfrac{\Delta B}{10}\]
Therefore,
\[\Delta B=0.01\]
The value of B can now be given as,
\[B=10\pm 0.01\]

So, the correct answer is “Option C”.

Note: Error can be defined as the difference between the actual value and calculated value of a certain quantity. There are three major types of errors namely, random error, logical error or systematic error. While calculating the actual value differs from original value. This difference is known as error value. These errors may arise due to a number of reasons such as human error, machine error or error in calculation.