To analyze the perimeter of the rectangle given by the expression \( 6w + 7 \), we can use the formula for the perimeter of a rectangle, which is \( P = 2l + 2w \), where \( l \) is the length and \( w \) is the width.
Given that the perimeter is \( 6w + 7 \), we can set this equal to the perimeter formula:
\[ 2l + 2w = 6w + 7 \]
To find \( l \), we can rearrange the terms:
\[ 2l = 6w + 7 - 2w \] \[ 2l = 4w + 7 \]
Dividing both sides by 2 to solve for \( l \):
\[ l = 2w + 3.5 \]
Now that we know the length in terms of the width, we can evaluate the statements:
-
The expression \( 13w \) also represents the perimeter of the rectangle and shows that the perimeter is 13 times the width.
- This is incorrect, as the perimeter is represented by \( 6w + 7 \), not \( 13w \).
-
The expression \( 2w+(4w+7) \) also represents the perimeter and shows that the length is 7 units more than four times the width.
- This expands to \( 2w + 4w + 7 = 6w + 7 \), but it incorrectly states the relationship about the length.
-
The expression \( 2(3w + 3.5) \) also represents the perimeter of the rectangle and shows that the perimeter is 3.5 units more than three times the width.
- This expands to \( 2(3w + 3.5) = 6w + 7 \), but it incorrectly describes the relationship.
-
The expression \( 2w + 2(2w + 3.5) \) also represents the perimeter and shows that the length is 3.5 units more than twice the width.
- This simplifies to \( 2w + 4w + 7 = 6w + 7 \). Moreover, from our earlier expression for \( l \), \( l = 2w + 3.5 \) indeed confirms that the length is 3.5 units more than twice the width.
The most accurate statement that correctly represents the perimeter and relationship between length and width is the last one:
The expression \( 2w + 2(2w + 3.5) \) also represents the perimeter and shows that the length is 3.5 units more than twice the width.
1 answer