https://file.aiquickdraw.com/mathbot/file/1730249099764suj694r3.jpg

To solve this problem, we need to use the concept of area of a trapezoid.

### Step-by-Step Solution

1. **Area of Trapezoid Formula**: The area \( A \) of a trapezoid is given by:
   \[
   A = \frac{1}{2} (b_1 + b_2) \cdot h
   \]
   where:
   - \( b_1 \) and \( b_2 \) are the lengths of the two parallel sides.
   - \( h \) is the height between the two parallel sides.

2. **Total Area**:
   - Given \( b_1 = 160 \) m and \( b_2 = 240 \) m with a height \( h = 40 \) m, we calculate the total area of the trapezoid:
   \[
   A_{\text{total}} = \frac{1}{2} \cdot (160 + 240) \cdot 40
   \]
   \[
   A_{\text{total}} = \frac{1}{2} \cdot 400 \cdot 40 = 8000 \, \text{m}^2
   \]

3. **Dividing the Area**:
   - Since the lot is divided into two equal areas, each section has an area of:
   \[
   A_{\text{half}} = \frac{8000}{2} = 4000 \, \text{m}^2
   \]

4. **Finding \( x \)**:
   - Now, we need to find the length \( x \) of the top section, which forms a smaller trapezoid with bases \( x \) and \( b_2 = 240 \) m, and height \( h = 20 \) m (since half of the height is 20 m).
   - Using the trapezoid area formula for the smaller section:
   \[
   4000 = \frac{1}{2} (x + 240) \cdot 20
   \]
   \[
   4000 = 10 (x + 240)
   \]
   \[
   400 = x + 240
   \]
   \[
   x = 400 - 240 = 160 \, \text{m}
   \]

The length \( x \) that divides the lot into two equal areas is **160 m**.

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Would you like more details or have any questions about the steps? 

### Related Questions

1. How can we verify the area calculation of the larger trapezoid?
2. What happens if we change the height of the sections?
3. Can this method be used for other irregularly shaped plots?
4. What if \( b_1 \) and \( b_2 \) were unknown—how would we solve for them?
5. How would the formula change if we divided the lot into three equal areas?

### Tip
For problems involving areas, identifying the type of shape (like trapezoids, triangles, etc.) and applying the appropriate formula is key to breaking down the problem.

The figure below shows the layout of a farm lot. Find the length x. Note: The lot is divided into two equal areas.

Learn how to determine the length x in a farm lot layout, given trapezoid dimensions with two equal areas. This problem involves applying the trapezoid area formula and is suitable for high school geometry students.

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