Question

Relative error in Z, given that Z = 1/A² is________

Answer 1

Relative error:

1. The ratio of the absolute error of a measurement to the measurement being taken.
2. This type of error is relative to the size of the item being measured.
3. The relative error is expressed as a percentage.
4. It has no units.
5. If the true measurement of the object is not known, then the relative error can be found using the measured value.

Step 1: Given information:

$Z=\frac{1}{A^2}$

Step 2: Formula to calculate relative error:

$$\begin{gathered} Relativeerror=\left(x_0-x\right)/x \\ =\left(∆x\right)/x \end{gathered}$$.

where

$x_0$= measured value

$x$= actual value

$∆x$= absolute error

Step 3: To find the relative error in Z.

We have $Z=\frac{1}{A^2}$

so now take log on both side

$\log Z=\log \left(\frac{1}{A^2}\right)$

Differentiating both sides with respect to $dZ$.

$$\begin{gathered} \frac{1}{Z}\times \frac{dZ}{dZ}=\frac{d\left(A^{-2}\right)}{dZ} \\ \frac{1}{Z}=\pm \left\{-2\times \frac{1}{A}\times \frac{dA}{dZ}\right\} \\ \frac{dZ}{Z}=\pm \left(-2\times \frac{dA}{A}\right) \\ \frac{dZ}{Z}=2\times \frac{dA}{A} \end{gathered}$$

Hence, the relative error in $Z$ is $\frac{dZ}{Z}$ is two times the relative error in $A$ that is $\frac{dA}{A}.$