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Difficulty: HardPerimeter and Area of 2D Figures

A piece of wire is bent to form a square with an area of 144144 square centimeters. If the same wire is straightened and then bent to form a rectangle whose length is 44 centimeters longer than its width, what is the area of this new rectangle?

  1. A
    4848 square centimeters
  2. 140140 square centimetersAnswer
  3. C
    144144 square centimeters
  4. D
    572572 square centimeters

Answer

140140 square centimeters
First, find the side length of the square. Since the area is 144144, the side length is 144=12\sqrt{144} = 12. The total length of the wire is the perimeter of the square, which is 12×4=4812 \times 4 = 48. This wire is then used to form a rectangle, so the rectangle's perimeter is also 4848. Let the width of the rectangle be WW. The length is W+4W + 4. The perimeter formula is 2(L+W)=P2(L + W) = P. Substituting the values gives 2(W+W+4)=482(W + W + 4) = 48, which simplifies to 4W+8=484W + 8 = 48. Solving for WW yields 4W=404W = 40, so W=10W = 10. The length is 10+4=1410 + 4 = 14. The area of the new rectangle is 10×14=14010 \times 14 = 140 square centimeters.

Step-by-Step Solution

1
Calculate the total length of the wire using the square's area.
The side of the square is 144=12\sqrt{144} = 12 cm. The perimeter is 12×4=4812 \times 4 = 48 cm.
The total length of the wire remains constant and equals the perimeter of both shapes.
2
Set up an equation for the rectangle's perimeter.
Let width be WW. Length is W+4W + 4. The perimeter is 2(W+(W+4))=482(W + (W + 4)) = 48.
The rectangle is made from the same wire, so its perimeter is 4848 cm, and its length is 44 cm longer than its width.
3
Solve for the width (WW) and length (LL) of the rectangle.
2(2W+4)=48⇒4W+8=48⇒4W=40⇒W=102(2W + 4) = 48 \Rightarrow 4W + 8 = 48 \Rightarrow 4W = 40 \Rightarrow W = 10 cm. The length is 10+4=1410 + 4 = 14 cm.
Solving the linear equation provides the dimensions needed to find the area.
4
Calculate the area of the rectangle.
Area = 10×14=14010 \times 14 = 140 square centimeters.
The area of a rectangle is its length multiplied by its width.

Key Concept

Perimeter and Area of 2D Figures
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Topics

Perimeter and Area of 2D Figures

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