The given problem describes a rectangular field with dimensions 25 m × 16.4 m, with two paths running through the center parallel to the sides. The widths of these paths are 1.7 m (longer side) and 2 m (shorter side).

Step 1: Calculate the Area of the Field

$\text{Area} = \text{Length} \times \text{Width} = 25 \times 16.4 = 410 \text{ m}^2$

Step 2: Calculate the Area of the Paths

• Longer Path:
  This path runs along the length (25 m) and has a width of 1.7 m.
  $\text{Area} = 25 \times 1.7 = 42.5 \text{ m}^2$

• Shorter Path:
  This path runs along the width (16.4 m) and has a width of 2 m.
  $\text{Area} = 16.4 \times 2 = 32.8 \text{ m}^2$

• Double-Counted Area (Intersection of Paths):
  Since both paths overlap at the center, this intersection is counted twice in the calculations above.
  The intersection area is a small rectangle of width 1.7 m and height 2 m: $\text{Area} = 1.7 \times 2 = 3.4 \text{ m}^2$

Thus, the total area of the paths is: $42.5 + 32.8 - 3.4 = 71.9 \text{ m}^2$

Step 3: Area of the Field Including Paths

$\text{Same as total field area: } 410 \text{ m}^2$

Step 4: Area of the Remaining Portion of the Field

$\text{Remaining Area} = 410 - 71.9 = 338.1 \text{ m}^2$

Final Answers:

1. The area of the paths is 71.9 m² (Option 1).
2. The total area of the field including paths is 410 m².
3. The area of the remaining portion of the field is 338.1 m² (Option 2).

Would you like further clarification or additional details?

Related Questions:

1. How would the calculation change if the paths were placed along the edges instead of the center?
2. What would be the area of the paths if their widths were doubled?
3. If the cost of maintaining the path is $5 per square meter, what would be the total cost?
4. How does the area of paths change if the paths are not perpendicular but diagonal?
5. What is the percentage of the total area occupied by the paths?

Tip:

When calculating areas involving paths, always subtract the overlapping portion to avoid overcounting.