The quantities x and y w that the relativeusing x√y to approximate x√y is given by . |∆x/x|+1/2|∆y/y|

Question

The quantities x and y has been estimated using X and Y with errors ∆xAnd ∆y respectively . show that the relative error in using x√y to approximate x√y is given by . |∆x/x|+1/2|∆y/y|

Given that x=2.5 and y=4.6 have been rounded off With corresponding percentage error of 4 and 5 . calculate the percentage error in x√y correct to 2 significant figure

Answer 1

Let $z = x\sqrt y$

Taking $\ln$ on both sides of the magnitudes,

$\ln|z| = \ln|x\sqrt y| = \ln|x| + \frac{1}{2}\ln|y|$

Taking differentials of both sides

$\frac{1}{|z|}|dz| = \frac{1}{|x|}|dx| + \frac{1}{2|y|}|dy|$

for small $x,($also $ y, z)$ $dx \approx \Delta x$

So, $\vert\frac{\Delta z}{z}\vert = \vert\frac{\Delta x}{x}\vert +\frac{1}{2}\vert\frac{\Delta y}{y}\vert$

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Now, $x = 2.5, y = 4.6, |\Delta x/x| = 4 \% = 0.04 , |\Delta y/y| = 5 \% = 0.05$, Use these values in the above formula and calculate the net percentage error of $z$