Which of the following describes the relationship between the length of a rectangle and its width as width varies and perimeter stays the same?
A. as width decreases, length stays constant
B. as width increases, length increases
C. as width decreases, length decreases
D. as width increases, length decreases
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To determine the relationship between the length of a rectangle and its width as the width varies and the perimeter remains constant, we can use the formula for the perimeter of a rectangle, which is given by:
Perimeter = 2(length + width)
Since the perimeter stays the same, we can set up an equation with the initial values of the length and width, and then compare them as the width changes.
Let's consider each option and evaluate it using the equation:
A. As width decreases, length stays constant:
If the width decreases, but the perimeter remains constant, the equation would be:
2(length + width_initial) = 2(length + width_decreased)
Simplifying:
length + width_initial = length + width_decreased
width_initial = width_decreased
This means that the initial width is equal to the decreased width, which contradicts the statement. So, option A is not correct.
B. As width increases, length increases:
If the width increases, we can write the equation as:
2(length + width_initial) = 2(length + width_increased)
Simplifying:
length + width_initial = length + width_increased
width_initial = width_increased
This shows that the initial width is equal to the increased width. Therefore, option B correctly describes the relationship.
C. As width decreases, length decreases:
If the width decreases, we can write the equation as:
2(length + width_initial) = 2(length + width_decreased)
Simplifying:
length + width_initial = length + width_decreased
width_initial = width_decreased
This shows that the initial width is equal to the decreased width. Therefore, option C correctly describes the relationship.
D. As width increases, length decreases:
If the width increases, we can write the equation as:
2(length + width_initial) = 2(length + width_increased)
Simplifying:
length + width_initial = length + width_increased
width_initial = width_increased
This means that the initial width is equal to the increased width, which contradicts the statement. So, option D is not correct.
In conclusion, the correct answer is B. As the width of a rectangle increases, the length also increases while keeping the perimeter constant. Option C, which states that as the width decreases, the length also decreases, is also a possible relationship.