In the problem, all four rectangles have the same area. To find which rectangle has the greatest perimeter, we first verify their areas and then calculate the perimeter for each.

Area Verification

All rectangles must have the same area. Let's calculate the area of each option:

• Option A: $1 \text{ foot} \times 12 \text{ feet} = 12 \text{ square feet}$
• Option B: $2 \text{ feet} \times 6 \text{ feet} = 12 \text{ square feet}$
• Option C: $3 \text{ feet} \times 4 \text{ feet} = 12 \text{ square feet}$
• Option D: $4 \text{ feet} \times 3 \text{ feet} = 12 \text{ square feet}$

Perimeter Calculation

The perimeter $P$ of a rectangle is calculated as $P = 2 \times (\text{length} + \text{width})$. Let's calculate each:

• Option A: $P = 2 \times (1 + 12) = 26 \text{ feet}$
• Option B: $P = 2 \times (2 + 6) = 16 \text{ feet}$
• Option C: $P = 2 \times (3 + 4) = 14 \text{ feet}$
• Option D: $P = 2 \times (4 + 3) = 14 \text{ feet}$

Thus, rectangle A, with a perimeter of 26 feet, has the greatest perimeter among the options.

Would you like any more details on this solution or have any other questions? Here are some follow-up topics you might consider:

1. Why does a more square-like rectangle (closer to equal sides) have a smaller perimeter for the same area?
2. How would the perimeter change if the area were different?
3. What would be the perimeter if one dimension is fixed and the other changes?
4. Could there be any practical applications of optimizing dimensions to minimize or maximize perimeter?
5. How would you calculate the dimensions of a rectangle if the perimeter and area are given?

Tip: Remember, for rectangles with the same area, the rectangle whose sides are closer in length will generally have a smaller perimeter. This is an example of the isoperimetric inequality in practical application.