The energy of a system is given as , where t is the time and . The error in the measurement of and t are 1.2% and 1.6%, respectively. At t = 5 s, maximum percentage error in the energy is [2025]

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Solution

Q.
The energy of a system is given as $E (t) = α^{3} e^{- β t}$ , where t is the time and $β = 0.3 s^{- 1}$ . The error in the measurement of $α$ and t are 1.2% and 1.6%, respectively. At t = 5 s, maximum percentage error in the energy is [2025]

1 4%

2 11.6%

3 6%

4 8.4%

Ans.
(3)

$d E = 3 α^{2} d α e^{- β t} + α^{3} e^{- β t} (- β) d t$

$\frac{d E}{E} = \frac{3 d α}{α} + (- β) d t$

$\frac{△ E}{E} = \frac{3 △ α}{α} + β △ t$

$(%) △ T = 1.6 \times 5 = 8 %$

$% (\frac{3 △ α}{α}) = 3 \times 1.2 % = 3.6 %$

Putting values we get,

$\frac{△ E}{E} % = 3.6 + 8 \times 0.3 = 6 %$ .

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