A physical quantity 'x'is calculated from the relation x=a^3b^2/root of cd. Calculate percentage error in x, if a, b, c and d are measured respectively with an error of 1%,3%,4% and 2%
Easy.
http://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm
error= 1*3+3*2-1/2*4-1/2*2=3+6-2-1=6percent error
To calculate the percentage error in x, we need to determine the effect of the errors in a, b, c, and d on the final result.
First, let's consider the effect of the error in a on x. The relative error in a, denoted as εa, can be calculated as:
εa = (error in a) / a = 1% / 100% = 0.01
Similarly, we can calculate the relative errors for b, c, and d as follows:
εb = (error in b) / b = 3% / 100% = 0.03
εc = (error in c) / c = 4% / 100% = 0.04
εd = (error in d) / d = 2% / 100% = 0.02
Now, let's express x using the calculated relative errors:
x = a^3 * b^2 / √(c * d)
Taking the natural logarithm of both sides of the equation:
ln(x) = ln(a^3 * b^2 / √(c * d))
Using the properties of logarithms, we can rewrite the equation as:
ln(x) = ln(a^3) + ln(b^2) - 0.5 * ln(c * d)
Now, let's differentiate both sides of the equation with respect to the variable a:
d(ln(x)) = 3 * (da / a) + 2 * (db / b) - 0.5 * (dc / c) - 0.5 * (dd / d)
The percentage error in x, denoted as δx, can be approximated using the formula:
δx ≈ |d(ln(x)) / ln(x)| * 100%
Substituting the relative errors we calculated earlier:
δx ≈ |(3 * εa + 2 * εb - 0.5 * εc - 0.5 * εd) / (ln(x))| * 100%
Let's calculate the percentage error using the given values:
δx ≈ |(3 * 0.01 + 2 * 0.03 - 0.5 * 0.04 - 0.5 * 0.02) / (ln(x))| * 100%