Question

A physical quantity $z$, depends upon two other physical quantities $x$ and $y$, as follows. $z=a{x}^2{y}^{1/2}$ where, a is a constant. In an experiment, the quantity $x$ is determined by measuring $z$ and $y$ and using the above expression. If the percentage of error in the measurement of $z$ and $y$ are $10\%$ and $12\%$ respectively, then the percentage of error in the determined value of $x$ is

• A. $2\%$
• B. $8\%$
• C. $15\%$
• D. without the value of the constant a, the percentage of error cannot be calculated

Answer 1

Answer: (A)

$z=a{x}^2{y}^{1/2}$
$\frac{\Delta z}{z}=2\frac{\Delta x}{x}+\frac{1}{2}\frac{\Delta y}{y}$
$\frac{\Delta z}{x}\times 100=\left[2\frac{\Delta x}{x}+\frac{1}{2}\frac{\Delta y}{y}\right]\times 100$
$10\%=2\frac{\Delta x}{x}\times 100+\frac{1}{2}\times 12\%$
$\frac{\Delta x}{2}\times 100=2\%$