26. Oil consumption on Pathfinder Island A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.



Day

Oil consumption rate (liters/hr)


Sun

0.019


Mon

0.020


Tue

0.021


Wed

0.023


Thu

0.025


Fri

0.028


Sat

0.031


Sun

0.035

To estimate the total amount of oil consumed by the diesel generator on Pathfinder Island over the week, we can use the **Trapezoidal Rule**. The Trapezoidal Rule is a numerical method used to approximate the integral (area under the curve), which in this case corresponds to the oil consumption rate over time.

### Trapezoidal Rule Formula:

\[
\text{Total oil consumed} \approx \frac{\Delta t}{2} \left( f_1 + 2f_2 + 2f_3 + \cdots + 2f_{n-1} + f_n \right)
\]

Where:
- \( \Delta t \) is the time interval between each day (which is 24 hours, since we're given daily consumption rates).
- \( f_1, f_2, \dots, f_n \) are the consumption rates at the different days.

### Data:
- \( f_1 = 0.019 \, \text{liters/hr} \)
- \( f_2 = 0.020 \, \text{liters/hr} \)
- \( f_3 = 0.021 \, \text{liters/hr} \)
- \( f_4 = 0.023 \, \text{liters/hr} \)
- \( f_5 = 0.025 \, \text{liters/hr} \)
- \( f_6 = 0.028 \, \text{liters/hr} \)
- \( f_7 = 0.031 \, \text{liters/hr} \)
- \( f_8 = 0.035 \, \text{liters/hr} \)

Each day corresponds to a 24-hour period, so \( \Delta t = 24 \, \text{hours} \).

### Applying the Trapezoidal Rule:
\[
\text{Total oil consumed} \approx \frac{24}{2} \left( 0.019 + 2(0.020 + 0.021 + 0.023 + 0.025 + 0.028 + 0.031) + 0.035 \right)
\]

Now, I'll calculate the total oil consumed.The estimated total amount of oil consumed by the diesel generator during the week, using the Trapezoidal Rule, is approximately **4.2 liters**.

Would you like further details on this calculation, or do you have any questions?

Here are five related questions for further exploration:
1. How would the result change if the time intervals between days were uneven?
2. Can the Simpson's Rule provide a better approximation in this case?
3. What if the oil consumption rate decreases over time instead of increasing?
4. How does the Trapezoidal Rule compare to calculating the exact integral?
5. How can you model the oil consumption rate as a continuous function for more accurate results?

**Tip:** The Trapezoidal Rule is generally more accurate when the data points form nearly straight lines between intervals.

Oil consumption on Pathfinder Island: A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.

This page explores how to estimate the total oil consumed by a diesel generator on Pathfinder Island using the Trapezoidal Rule. With given daily oil consumption rates over a week, the total amount of oil is calculated using numerical integration techniques. Suitable for Grades 10-12.

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