Question

The power consumption readings (in watt) by an instrument at fixed intervals of time (in seconds) are tabulated below:  

\[ \begin{array}{|c|c|} \hline \text{Time (s)} & \text{Power consumption (W)} \\ \hline 0.0 & 8.6 \\ 0.6 & 9.2 \\ 1.2 & 7.8 \\ 1.8 & 6.4 \\ 2.4 & 7.2 \\ 3.0 & 8.6 \\ 3.6 & 11.2 \\ \hline \end{array} \] 
Using Simpson's 1/3rd rule, the energy expenditure of the instrument in joules is _____.[Round off to two decimal places]

Hint:

Simpson's 1/3rd rule is a useful method for approximating the integral of a function, especially for evenly spaced data points.

Answer 1

Correct Answer: 29.2

Simpson's 1/3rd rule for numerical integration is given by: \[ E = \frac{\Delta t}{3} \left( P_0 + 4 \sum_{\text{odd}} P_i + 2 \sum_{\text{even}} P_j + P_n \right) \] where: - \( P_i \) is the power consumption at time \( t_i \), - \( \Delta t = 0.6 \) is the interval between the times, - The sums are taken over odd and even indices for the power values. Substitute the given values into the equation: \[ E = \frac{0.6}{3} \left( 8.6 + 4(9.2 + 6.4 + 7.2) + 2(7.8 + 8.6) + 11.2 \right) \] \[ E = 0.2 \times \left( 8.6 + 4(22.8) + 2(16.4) + 11.2 \right) \] \[ E = 0.2 \times \left( 8.6 + 91.2 + 32.8 + 11.2 \right) = 0.2 \times 143.8 = 28.76 \, \text{joules}. \] Thus, the energy expenditure is \( \boxed{28.76} \, \text{joules} \) (rounded to two decimal places).