Question

ABCD is a rectangle of dimensions 80 cm \(\times\) 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

Hint:

Always double-check your final values against physical constraints.
The margin \(x\) cannot exceed half of the shortest side of the rectangle. Since the shortest side is 60 cm, \(x\) must be strictly less than 30 cm. This helps you instantly rule out the root \(x = 60\) without wasting time.

Answer 1

Correct Answer: 10

Step 1: Understanding the Question:
We are given an outer rectangle \(ABCD\) of dimensions \(80\text{ cm} \times 60\text{ cm}\).
An inner rectangle \(PQRS\) is drawn leaving a uniform margin of width \(x\text{ cm}\) along all four edges.
The area of the inner rectangle \(PQRS\) is exactly half of the area of the outer rectangle \(ABCD\).
We need to set up a quadratic equation and solve for the value of \(x\).

Step 2: Key Formula or Approach: 1. \(\text{Area of a rectangle} = \text{Length} \times \text{Width}\).
2. Dimensions of outer rectangle \(ABCD\):
- \(\text{Length} = 80\text{ cm}\)
- \(\text{Width} = 60\text{ cm}\)
3. Dimensions of inner rectangle \(PQRS\):
- \(\text{Length} = 80 - 2x\) (subtracting margins of \(x\) on both left and right sides)
- \(\text{Width} = 60 - 2x\) (subtracting margins of \(x\) on both top and bottom sides)
4. The given condition is:
\[ \text{Area of } PQRS = \frac{1}{2} \times \text{Area of } ABCD \]

Step 3: Detailed Explanation:

• Calculate the area of the outer rectangle \(ABCD\):
\[ \text{Area}(ABCD) = 80 \times 60 = 4800\text{ cm}^2 \]

• According to the given condition, calculate the target area for \(PQRS\):
\[ \text{Area}(PQRS) = \frac{1}{2} \times 4800 = 2400\text{ cm}^2 \]

• Express the area of \(PQRS\) in terms of \(x\):
\[ (80 - 2x)(60 - 2x) = 2400 \]

• Expand the left-hand side of the equation:
\[ 4800 - 160x - 120x + 4x^2 = 2400 \] \[ 4x^2 - 280x + 4800 = 2400 \]

• Move all terms to one side to set the quadratic equation to zero:
\[ 4x^2 - 280x + 2400 = 0 \]

• Divide the entire equation by 4 to simplify:
\[ x^2 - 70x + 600 = 0 \]

• Solve this quadratic equation by factoring:
- We need two numbers that multiply to \(600\) and add up to \(-70\). These are \(-60\) and \(-10\).
\[ x^2 - 60x - 10x + 600 = 0 \] \[ x(x - 60) - 10(x - 60) = 0 \] \[ (x - 10)(x - 60) = 0 \] - This gives two possible mathematical solutions: \(x = 10\) or \(x = 60\).

• Analyze the physical feasibility of the solutions:
- If \(x = 60\text{ cm}\), then the width of the inner rectangle would be \(60 - 2(60) = -60\text{ cm}\), which is physically impossible.
- Therefore, we discard \(x = 60\).
- Hence, \(x = 10\text{ cm}\).


Step 4: Final Answer:
The value of \(x\) is 10 cm.