We will apply Simpson’s Rule to estimate the energy used during the time period. Simpson's Rule is defined as:

$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + f(x_n)\right],$

where:

• $n$ (the number of subintervals) must be even,
• $\Delta x = \frac{b - a}{n}$,
• and the coefficients alternate between $4$ and $2$, with $1$ at both ends.

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Step 1: Key Information

• Time is from $0:00$ to $6:00$, so $a = 0$ and $b = 6$ hours.
• There are 13 data points (at 30-minute intervals), so $n = 12$ subintervals (even as required).
• $\Delta x = \frac{6 - 0}{12} = 0.5$ hours.

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Step 2: Assign Power Values to $P(x)$

The power values correspond to $f(x)$, as follows: $P(x) = [1814, 1735, 1686, 1646, 1637, 1609, 1604, 1611, 1621, 1666, 1745, 1886, 2052].$

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Step 3: Apply Simpson’s Rule

Using the formula: $\int_{0}^{6} P(t) \, dt \approx \frac{\Delta x}{3} \left[P_0 + 4P_1 + 2P_2 + 4P_3 + \cdots + 2P_{n-2} + 4P_{n-1} + P_n\right],$ we substitute the power values: $\int_{0}^{6} P(t) \, dt \approx \frac{0.5}{3} \Big[1814 + 4(1735) + 2(1686) + 4(1646) + 2(1637) + 4(1609) + 2(1604) + 4(1611) + 2(1621) + 4(1666) + 2(1745) + 4(1886) + 2052\Big].$

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Step 4: Simplify the Expression

Evaluate each term: [ \begin{aligned} \int_{0}^{6} P(t) , dt &\approx \frac{0.5}{3} \Big[1814 + 6940 + 3372 + 6584 + 3274 + 6436 + 3208 + 6444 + 3242 + 6664 + 3490 + 7544 + 2052\Big] \ &\approx \frac{0.5}{3} \Big[59764\Big]. ]

Now compute: $\int_{0}^{6} P(t) \, dt \approx \frac{0.5}{3} \times 59764 = \frac{29882}{3} \approx 9960.67 \, \text{megawatt-hours}.$

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Final Answer:

The energy used during this time period is approximately: $\boxed{9960.67 \, \text{megawatt-hours}}.$

Would you like more explanation or a breakdown of any step? 😊

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Questions:

1. What is Simpson’s Rule, and why is it accurate for numerical integration?
2. How does the step size ($\Delta x$) affect the accuracy of Simpson’s Rule?
3. What is the physical relationship between power and energy in this problem?
4. How would the result change if $n$ were not even? Could we still apply Simpson’s Rule?
5. How can we use trapezoidal rule to solve this problem, and how does it compare to Simpson’s Rule?

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Tip:

Simpson’s Rule requires the function to be smooth and well-behaved; it’s particularly accurate for functions resembling parabolas over small intervals.