Connor took three measurements at 6m intervals across an irregular-shaped ﬁeld. The
measurements were 10 m, 12 m and 8m. Use two applications of the trapezoidal rule to
calculate the area of the ﬁeld. Answer correct to the nearest square metre.

To calculate the area of the irregular-shaped field using the trapezoidal rule, we can break the problem down into two applications of the rule, based on the three given measurements.

### Trapezoidal Rule Formula:
The area \( A \) between two points using the trapezoidal rule is given by:

\[
A = \frac{h}{2} \times (y_1 + y_2)
\]

where:
- \( h \) is the distance between the points (in this case, 6 meters),
- \( y_1 \) and \( y_2 \) are the two consecutive measurements (heights of the trapezoid).

We will apply the trapezoidal rule twice—once for the first and second measurements, and once for the second and third measurements.

### Step 1: First application (between the first two measurements)
The first two measurements are 10 m and 12 m, and the distance between them is 6 meters.

\[
A_1 = \frac{6}{2} \times (10 + 12) = 3 \times 22 = 66 \, \text{m}^2
\]

### Step 2: Second application (between the second and third measurements)
The second and third measurements are 12 m and 8 m, and the distance between them is also 6 meters.

\[
A_2 = \frac{6}{2} \times (12 + 8) = 3 \times 20 = 60 \, \text{m}^2
\]

### Step 3: Total area
Now, we add the two areas to get the total area of the field.

\[
\text{Total Area} = A_1 + A_2 = 66 + 60 = 126 \, \text{m}^2
\]

Thus, the area of the field is approximately **126 square meters**.

---

Would you like a more detailed explanation of any step, or have any questions? 

Here are 5 related questions:
1. How would the trapezoidal rule work if the distances between the measurements were not equal?
2. What is the significance of applying the trapezoidal rule multiple times for irregular shapes?
3. How accurate is the trapezoidal rule compared to other methods for estimating areas of irregular fields?
4. How does increasing the number of measurements affect the accuracy of the trapezoidal rule?
5. Can the trapezoidal rule be used in 3D to calculate volumes instead of areas?

**Tip**: The more subdivisions (or intervals) you use, the more accurate your trapezoidal rule approximation becomes.

Connor took three measurements at 6m intervals across an irregular-shaped field. The measurements were 10 m, 12 m, and 8 m. Use two applications of the trapezoidal rule to calculate the area of the field. Answer correct to the nearest square metre.

Learn how to use the trapezoidal rule to calculate the area of an irregular field with measurements at 6m intervals. This problem demonstrates numerical integration with two applications of the trapezoidal rule, providing a step-by-step solution. Suitable for students in Grades 9-11.

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