Danica has laid out floor tiles so they from a rectangle with a perimeter of 18 inches what is the difference between the greatest and least possible areas of the rectangle
Answer (2)
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.
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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