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The problem asks to estimate the area of the paddock, which has an irregular boundary, using the trapezoidal rule. Here's how we can approach this problem:

### Steps for solving using the trapezoidal rule:

1. The paddock is divided into two sections, so we will apply the trapezoidal rule twice.
2. The distances between the boundary lines are given: 140 m. The heights of the irregular boundary are:
   - \( h_1 = 110 \, m \)
   - \( h_2 = 98 \, m \)
   - \( h_3 = 120 \, m \)
   
   These heights correspond to the two sections for which we will apply the trapezoidal rule.

#### Trapezoidal Rule Formula:
For one section:
\[
\text{Area} = \frac{1}{2} \times ( \text{Base1} + \text{Base2} ) \times \text{Height}
\]

We'll apply the rule to two trapezoids:

1. **First trapezoid** (between \( h_1 = 110 \, m \) and \( h_2 = 98 \, m \)):
   \[
   \text{Area}_1 = \frac{1}{2} \times (110 + 98) \times \frac{140}{2}
   \]

2. **Second trapezoid** (between \( h_2 = 98 \, m \) and \( h_3 = 120 \, m \)):
   \[
   \text{Area}_2 = \frac{1}{2} \times (98 + 120) \times \frac{140}{2}
   \]

Next, I will calculate the area of both trapezoids to estimate the total area of the paddock.The estimated area of the paddock using the trapezoidal rule is approximately **14,910 square meters**.

Would you like further clarification or details on any part of the calculation? Here are some related questions for practice:

1. How would the area change if one of the heights was increased by 10 meters?
2. Can the trapezoidal rule be applied to more than two sections for better accuracy?
3. How does the trapezoidal rule approximate areas under irregular boundaries compared to other methods like Simpson's rule?
4. What assumptions are made about the shape of the paddock in applying the trapezoidal rule?
5. How can we verify the accuracy of this estimation using another method?

**Tip:** The more trapezoids you use in the trapezoidal rule, the more accurate your area estimation will be for irregular boundaries.

Below is a paddock that has an irregular boundary. Using two applications of the trapezoidal rule, estimate the area of the paddock.

Learn how to estimate the area of a paddock with an irregular boundary using the trapezoidal rule. This geometry problem involves applying the formula for the area of a trapezoid to estimate the total area in two sections. Suitable for high school students.

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