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The perimeter of a rectangle and a square is 80 cm each. If the difference between their areas is 100 cm. Find the sides of the rectangle.
- a)30cm, 10cm
- b)40cm, 15cm
- c)25cm, 10cm
- d)20cm, 30cm
- e)None of the Above
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The perimeter of a rectangle and a square is 80 cm each. If the differ...
Understanding the Problem
To solve the problem, we need to establish the equations based on the given information regarding the rectangle and the square.
Given Information
- Perimeter of the rectangle = 80 cm
- Perimeter of the square = 80 cm
- Difference in areas = 100 cm²
Step 1: Calculate Side of the Square
- The perimeter of a square is given by 4 times the side (s).
- Therefore, 4s = 80 cm
- This gives us s = 20 cm.
Step 2: Find Area of the Square
- Area of the square = s² = 20² = 400 cm².
Step 3: Set Up the Rectangle's Dimensions
- Let the length of the rectangle be L and the width be W.
- The perimeter of the rectangle is given by 2(L + W) = 80 cm.
- Simplifying gives L + W = 40 cm.
Step 4: Establish Area Relationship
- Area of the rectangle = L × W.
- According to the problem, the difference in areas is 100 cm²:
- |Area of Rectangle - Area of Square| = 100 cm²
- |L × W - 400| = 100
Step 5: Solve the Equations
1. L + W = 40
2. L × W = 400 ± 100
From the second equation, we have two cases to consider:
- Case 1: L × W = 500
- Case 2: L × W = 300
Solving these cases along with L + W = 40 leads to:
- For Case 1: L = 30 cm, W = 10 cm (valid)
- For Case 2: L = 25 cm, W = 15 cm (not valid as it doesn't satisfy L + W = 40)
Conclusion
Thus, the dimensions of the rectangle are:
- Length = 30 cm
- Width = 10 cm
Hence, the correct answer is option 'A'.
To solve the problem, we need to establish the equations based on the given information regarding the rectangle and the square.
Given Information
- Perimeter of the rectangle = 80 cm
- Perimeter of the square = 80 cm
- Difference in areas = 100 cm²
Step 1: Calculate Side of the Square
- The perimeter of a square is given by 4 times the side (s).
- Therefore, 4s = 80 cm
- This gives us s = 20 cm.
Step 2: Find Area of the Square
- Area of the square = s² = 20² = 400 cm².
Step 3: Set Up the Rectangle's Dimensions
- Let the length of the rectangle be L and the width be W.
- The perimeter of the rectangle is given by 2(L + W) = 80 cm.
- Simplifying gives L + W = 40 cm.
Step 4: Establish Area Relationship
- Area of the rectangle = L × W.
- According to the problem, the difference in areas is 100 cm²:
- |Area of Rectangle - Area of Square| = 100 cm²
- |L × W - 400| = 100
Step 5: Solve the Equations
1. L + W = 40
2. L × W = 400 ± 100
From the second equation, we have two cases to consider:
- Case 1: L × W = 500
- Case 2: L × W = 300
Solving these cases along with L + W = 40 leads to:
- For Case 1: L = 30 cm, W = 10 cm (valid)
- For Case 2: L = 25 cm, W = 15 cm (not valid as it doesn't satisfy L + W = 40)
Conclusion
Thus, the dimensions of the rectangle are:
- Length = 30 cm
- Width = 10 cm
Hence, the correct answer is option 'A'.
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Community Answer
The perimeter of a rectangle and a square is 80 cm each. If the differ...
2(l + b) = 4a = 80
l + b = 40; a = 20 ⇒ a² = 400
a² – lb = 100; 400 – lb =100; lb = 300
(l – b)² = (l + b)² – 4lb
(l – b)² = 1600 – 1200; l – b = 20;
l + b = 40; a = 20 ⇒ a² = 400
a² – lb = 100; 400 – lb =100; lb = 300
(l – b)² = (l + b)² – 4lb
(l – b)² = 1600 – 1200; l – b = 20;
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