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Step-by-Step Solution
Step 1: Write down the given expression for z
We are given
$z = \frac{a^2 \, b^{\frac{2}{3}}}{\sqrt{c}\, d^3}$.
Step 2: Express the fractional (relative) error in z
For a general quantity
$z = a^p\, b^q\, c^r\, d^s$,
the relative error can be written as
$\frac{\Delta z}{z} = \left|p\right|\frac{\Delta a}{a} + \left|q\right|\frac{\Delta b}{b} + \left|r\right|\frac{\Delta c}{c} + \left|s\right|\frac{\Delta d}{d}$.
In our case,
$p = 2$,
$q = \frac{2}{3}$,
$r = -\frac{1}{2}$ (since in the denominator it becomes negative power),
and
$s = -3$.
So, the fractional (relative) error in z is
$\frac{\Delta z}{z} = 2 \frac{\Delta a}{a} + \frac{2}{3} \frac{\Delta b}{b} + \left|\,-\frac{1}{2}\right|\frac{\Delta c}{c} + \left|\,-3\right|\frac{\Delta d}{d}.$
Step 3: Convert the fractional errors into percentage errors
We are given the percentage errors (denoted as % error) in each quantity as follows:
% error in a = 2%
% error in b = 1.5%
% error in c = 4%
% error in d = 2.5%
Hence,
$\frac{\Delta a}{a} \times 100 = 2\%$,
$\frac{\Delta b}{b} \times 100 = 1.5\%$,
$\frac{\Delta c}{c} \times 100 = 4\%$,
$\frac{\Delta d}{d} \times 100 = 2.5\%$.
Step 4: Substitute and compute the total percentage error
Substitute these values in the expression:
$\left(\frac{\Delta z}{z}\right) \times 100 = \left(2 \times 2\% + \frac{2}{3} \times 1.5\% + \frac{1}{2} \times 4\% + 3 \times 2.5\%\right).$
Evaluating each term:
$2 \times 2\% = 4\%$
$\frac{2}{3} \times 1.5\% = 1\%$
$\frac{1}{2} \times 4\% = 2\%$
$3 \times 2.5\% = 7.5\%$
Adding them all together:
Total percentage error
$= 4\% + 1\% + 2\% + 7.5\% = 14.5\%.$
Step 5: Final Answer
Therefore, the percentage error in z is 14.5%.
