A wire of length 20 cm is to be bent into a rectangle. Which of the following statements is/are correct? I. The rectangle of the largest area is the square. Select the answer using the code given below.
II. It is possible to form a rectangle of an area of \(27 \, \text{cm}^2\).
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To determine which statement(s) regarding the wire bent into a rectangle is/are correct, we need to go through a systematic analysis:
- Understanding the Problem: We have a wire of length 20 cm, which is to be bent into a rectangle. We need to assess the truth of two statements regarding the resultant rectangle:
- I. The rectangle of the largest area is the square.
- II. It is possible to form a rectangle with an area of \(27 \, \text{cm}^2\).
- Analyzing Statement I:
- To maximize the area of a rectangle with a fixed perimeter, the rectangle should be a square. This is a derived fact from optimization where for a given perimeter, a square encloses the maximum area.
- The perimeter of a square with side length \(s\) is \(4s\). Setting this equal to 20 cm gives:
- The area of the square is \(s^2 = 5^2 = 25 \, \text{cm}^2\).
- This confirms that Statement I is correct, as no rectangle of the same perimeter can have an area larger than 25 cm2.
- Analyzing Statement II:
- Let's denote the length of the rectangle as \(l\) and width as \(w\).
- We know the perimeter is given by \(2l + 2w = 20\) which simplifies to \(l + w = 10\).
- The area \(A\) of the rectangle is \(lw\). We need this area to be 27 cm2.
- Substitute \(w = 10 - l\) into the area equation:
- The discriminant of this quadratic equation is \(b^2 - 4ac = (-10)^2 - 4 \times 1 \times 27 = 100 - 108 = -8\).
- The negative discriminant indicates that no real solution exists for \(l\) and \(w\) that would satisfy the equation. Thus, Statement II is incorrect.
- Conclusion: Only Statement I is correct, as the largest area for the given perimeter configuration is indeed a square of 25 cm2. Statement II is not feasible due to the negative discriminant in the quadratic equation, showing no real rectangle can have an area of 27 cm2 with the given perimeter.
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