A quantity $ Q $ is formulated as $ Q = X^{-2} Y^{3/2} Z^{-2/5} $. $ X $, $ Y $, and $ Z $ are independent parameters which have fractional errors of 0.1, 0.2, and 0.5, respectively in measurement. The maximum fractional error of $ Q $ is:

Question

The energy of a system is given as $ E(t) = \alpha e^{-\beta t $, where $ t $ is the time and $ \beta = 0.3 \, \text{s}^{-1} $. The errors in the measurement of $ \alpha $ and $ t $ are 1.2 percent and 1.6 percent, respectively. At $ t = 5 $ s, the maximum percentage error in the energy is:}

Answer 1

The maximum fractional error in the quantity $Q$ is determined by propagating the errors from parameters $X$, $Y$, and $Z$ through the equation $Q = X^{-2} Y^{3/2} Z^{-2/5}$. For a quantity of the form $Q = X^a Y^b Z^c$, the fractional error is given by $\frac{\Delta Q}{Q} = |a| \frac{\Delta X}{X} + |b| \frac{\Delta Y}{Y} + |c| \frac{\Delta Z}{Z}$.

For the given function, $a = -2$, $b = \frac{3}{2}$, and $c = -\frac{2}{5}$. The fractional errors are:

$\frac{\Delta X}{X} = 0.1$
$\frac{\Delta Y}{Y} = 0.2$
$\frac{\Delta Z}{Z} = 0.5$

Substituting these values into the fractional error formula:

$\frac{\Delta Q}{Q} = |-2| \times 0.1 + \left|\frac{3}{2}\right| \times 0.2 + \left|-\frac{2}{5}\right| \times 0.5$

The individual terms are:

$|-2| \times 0.1 = 0.2$
$\left|\frac{3}{2}\right| \times 0.2 = 0.3$
$\left|-\frac{2}{5}\right| \times 0.5 = 0.2$

Summing these terms yields:

$\frac{\Delta Q}{Q} = 0.2 + 0.3 + 0.2 = 0.7$

The maximum fractional error of $Q$ is 0.7.

A quantity $ Q $ is formulated as $ Q = X^{-2} Y^{3/2} Z^{-2/5} $. $ X $, $ Y $, and $ Z $ are independent parameters which have fractional errors of 0.1, 0.2, and 0.5, respectively in measurement. The maximum fractional error of $ Q $ is:

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The Correct Option is A

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