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the difference between the greatest and least possible areas of the rectangle is 11.25 square inches.
The perimeter P of a rectangle is given by the formula P = 2 l + 2 w , where l is the length and w is the width of the rectangle. For a given perimeter, the greatest possible area of a rectangle is achieved when the rectangle is a square (since the square is the rectangle with the largest area for a given perimeter).
For a square, the perimeter would be P = 4 s , where s[/tex] is the side length of the square. To find the side length of the square, we would set [tex]\( 4s = 18 and solve for s , and the area A would be s 2 .
The least possible area of a rectangle for a given perimeter would theoretically be as close to zero as possible, which would happen when the length is very large and the width is very small (approaching a line). In practical terms, this isn't quite possible, so we usually consider the least possible area to be when the rectangle has dimensions that are non-zero but differ greatly, such as when one side is 1 inch and the other is P /2 − 1 .
Let's calculate both:

Greatest area (square):

4 s = 18
s = 4 18 ​ = 4.5
Area greatest ​ = s 2 = ( 4.5 ) 2

Least area (rectangle):

If one side is 1 inch, the other side would be 2 P ​ − 1 = 2 18 ​ − 1 = 9 − 1 = 8 inches.
Area least ​ = 1 × 8 = 8
The difference between the greatest and least possible areas would be
\( \text{Area}_\text{greatest} - \text{Area}_\text{least} \).
Let's calculate these areas and their difference.
The greatest possible area, which is when the rectangle is a square with a side length of 4.5 inches, is:
Area greatest ​ = ( 4.5 ) 2 = 20.25 square inches
The least possible area for practical purposes, assuming a width of 1 inch, is:
Area least ​ = 1 × 9 = 9 square inches
The difference between the greatest and least possible areas is:
Area difference ​ = 20.25 − 9 = 11.25 square inches
Thus, the difference between the greatest and least possible areas of the rectangle is 11.25 square inches.

Answered by asad00ira | 2024-06-18

The difference between the greatest and least possible areas of a rectangle with a perimeter of 18 inches is 12.25 square inches. The greatest area is achieved when the rectangle is a square, giving an area of 20.25 square inches. The least area occurs with one side significantly larger than the other, resulting in an area of 8 square inches.
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Answered by asad00ira | 2024-12-17

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