Final answer:
Given a perimeter of 500 feet, the area of a square would be 15,625 square feet, which is smaller than 21,780 square feet. Therefore, lots with a perimeter of 500 feet are smaller in size compared to lots with an area of 21,780 square feet.
Step-by-step explanation:
The question is asking whether lots with a given perimeter will be the same, larger, or smaller in size compared to lots with a given area. To answer this, we can use mathematical relationships between the perimeters and areas of shapes. For instance, consider the case of a square. If a square has a perimeter of 500 feet, its sides would each be 125 feet long (500 divided by 4). The area of this square would be 125 feet times 125 feet, which is 15,625 square feet. Now, if we are comparing this to lots with an area of 21,780 square feet, we can conclude that the lots with a perimeter of 500 feet would be smaller in size since 15,625 square feet is less than 21,780 square feet.
We can also infer that if we were to maintain a perimeter of 500 feet with other shapes (like rectangles or circles), the areas would still be less than 21,780 square feet given that the maximum area enclosed by a given perimeter is that of a circle (as per the isoperimetric inequality). Thus, regardless of the shape, if the perimeter is fixed at 500 feet, the area will be less than a square of the same perimeter, which is already smaller than 21,780 square feet.
Therefore, the correct answer is A. The lots will be smaller in size.