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:}

• A. 4%
• B. 11.6%
• C. 6%
• D. 8.4%

Hint:

When calculating the percentage error in exponential functions, remember to account for the contributions from each variable and their derivatives.

Answer 1

The Correct Option is C

The energy of the system is given by: \[ E(t) = \alpha e^{-\beta t}. \] The percentage error in \( E \) is the sum of the percentage errors in \( \alpha \) and \( t \), weighted by the partial derivatives of \( E \) with respect to \( \alpha \) and \( t \). The percentage error in \( E \) is: \[ \% \Delta E = \% \Delta \alpha + \% \Delta t \cdot \left( -\beta \cdot t \right). \] Substitute the given values and calculate the error at \( t = 5 \) s: \[ \% \Delta E = 1.2 + 1.6 \cdot (-0.3 \times 5) = 1.2 + (-2.4) = 6%. \]