The energy of a system is given as $\mathrm{E}(\mathrm{t})=\alpha^3 \mathrm{e}^{-\beta t}$, where t is the time and $\beta=0.3 \mathrm{~s}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2 \%$ and $1.6 \%$, respectively. At $t=5 \mathrm{~s}$, maximum percentage error in the energy…

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The energy of a system is given as $\mathrm{E}(\mathrm{t})=\alpha^3 \mathrm{e}^{-\beta t}$, where t is the time and $\beta=0.3 \mathrm{~s}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2 \%$ and $1.6 \%$, respectively. At $t=5 \mathrm{~s}$, maximum percentage error in the energy is :

Aicharya · Accepted Answer

$\begin{aligned} & \mathrm{E}=\alpha^3 \mathrm{e}^{-\beta t} \ & \ln \mathrm{E}=3 \ln \alpha-\beta \mathrm{t} \ & \left(\frac{\mathrm{dE}}{\mathrm{E}}\right)_{\max }=\frac{3 \mathrm{~d} \alpha}{\alpha}+\beta \frac{\mathrm{dt}}{\mathrm{t}} \times \mathrm{t} \ & =3 \times 1.2 \%+(0.3 \times 1.6 \times 5) \% \ & =6 \%\end{aligned}$

The energy of a system is given as $\mathrm{E}(\mathrm{t})=\alpha^3 \mathrm{e}^{-\beta t}$, where t is the…

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