The problem in the image involves designing a rectangular corral and calculating the amount of fencing required. Here's the breakdown:

1. Initial Details:

   • The area of the corral is 576 square meters.
   • The given dimensions are 12 meters by 48 meters (which indeed provides $12 \times 48 = 576 \, \text{m}^2$).

2. Perimeter Calculation: The perimeter of a rectangle is given by: $P = 2(\text{length} + \text{width})$ Substituting the dimensions: $P = 2(48 + 12) = 2(60) = 120 \, \text{meters}$ Hence, the rancher needs 120 meters of fencing for this design.

3. Economical Solution: The most economical solution minimizes the perimeter while maintaining the area of 576 $\text{m}^2$. This occurs when the corral is as close to a square as possible, as a square minimizes perimeter for a given area.

   • If the corral were square, the side length $s$ would satisfy: $s^2 = 576 \quad \Rightarrow \quad s = \sqrt{576} = 24 \, \text{meters}$
   • The perimeter for a square corral would be: $P = 4s = 4(24) = 96 \, \text{meters}$ This is less fencing than the rectangular design (120 meters). Thus, a square corral is the most economical solution.

Final Answer:

• The rancher needs 120 meters of fence for the given design (12 by 48 meters).
• The rancher has not found the most economical solution; a square corral requires only 96 meters of fence.

Would you like me to explore or explain any aspect further?

Here are 5 related questions to expand your understanding:

1. How is the perimeter formula derived for any rectangle?
2. Why does a square minimize the perimeter for a given area?
3. Can you calculate the perimeter for other rectangular designs with an area of 576 $\text{m}^2$?
4. How would the perimeter change if the length and width of the rectangle were swapped?
5. Why is it important to minimize the perimeter in real-world fencing problems?

Tip: When designing enclosures with fixed areas, aim for shapes with smaller perimeters to save on fencing material!