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Standard Deviation

The energy of...

Question

The energy of a system as a function of time $t$ is given as $E = A^{2} \times e^{- α t}$ , where $α = 0.2 s^{- 1}$ . The measurement of $A$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $E (t)$ at $t = 5 s$ is

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Solution

$E (t) = A^{2} e^{- α t}$
and $α = 0.2 / s$

We have, $\frac{d A}{A} = 1.25$

$\frac{d t}{t} = 1.5$ .

Now, $l o g E = 2 l o g A - α t$ . ..... (I)

Now diffrentiating the equation (I)
$\frac{d E}{E} = 2 \frac{d A}{A} - α d t$ .
Hence for calculating maximum error
Or, $(\frac{d E}{E})_{m a x} = \pm 2 \frac{d A}{A} \pm α d t$
$= \pm 2 (1.25) \pm 0.2 (7.5)$
$= \pm 2 (1.25) \pm 1.5$
$= \pm 4 %$

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