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The perimeter of a rectangle and a square is 160 cm each. If the difference between their areas is 500 cm. If the area of the rectangle is less than that of a Square then find the area of the rectangle?
- a)1800 cm²
- b)1500 cm²
- c)1100 cm²
- d)1600 cm²
- e)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The perimeter of a rectangle and a square is 160 cm each. If the diffe...
Perimeter of rectangle = Perimeter of Square = 160
4a = 160 ⇒ a = 40 Area of square = 1600
1600 – lb = 500
lb = 1100 cm²
4a = 160 ⇒ a = 40 Area of square = 1600
1600 – lb = 500
lb = 1100 cm²
Most Upvoted Answer
The perimeter of a rectangle and a square is 160 cm each. If the diffe...
Understanding the Problem
We have a rectangle and a square, both with a perimeter of 160 cm. The difference in their areas is 500 cm², and the area of the rectangle is less than that of the square.
Step 1: Finding the Side Lengths
- For the Square:
- Perimeter = 4 × side
- 160 = 4 × side
- Side of square = 160 / 4 = 40 cm
- For the Rectangle:
- Let length = L and width = W.
- Perimeter = 2(L + W) = 160
- L + W = 80
Step 2: Area Calculation
- Area of the Square:
- Area = side² = 40² = 1600 cm²
- Area of the Rectangle:
- Area = L × W
- Given: Area of Square - Area of Rectangle = 500 cm²
- Thus, 1600 - L × W = 500
- L × W = 1600 - 500 = 1100 cm²
Step 3: Solving for Length and Width
- We have two equations:
1. L + W = 80
2. L × W = 1100
Using the first equation to express W in terms of L:
- W = 80 - L
Substituting into the second equation:
- L(80 - L) = 1100
- 80L - L² = 1100
- L² - 80L + 1100 = 0
Step 4: Finding Roots
Now, we can apply the quadratic formula:
- L = [80 ± √(80² - 4 × 1 × 1100)] / (2 × 1)
- L = [80 ± √(6400 - 4400)] / 2
- L = [80 ± √2000] / 2
- L = [80 ± 20√5] / 2
- L = 40 ± 10√5
Thus, we can find W:
- W = 80 - L
- The area of the rectangle, L × W, is confirmed to be 1100 cm².
Conclusion
The area of the rectangle is indeed 1100 cm², which corresponds to option 'C'.
We have a rectangle and a square, both with a perimeter of 160 cm. The difference in their areas is 500 cm², and the area of the rectangle is less than that of the square.
Step 1: Finding the Side Lengths
- For the Square:
- Perimeter = 4 × side
- 160 = 4 × side
- Side of square = 160 / 4 = 40 cm
- For the Rectangle:
- Let length = L and width = W.
- Perimeter = 2(L + W) = 160
- L + W = 80
Step 2: Area Calculation
- Area of the Square:
- Area = side² = 40² = 1600 cm²
- Area of the Rectangle:
- Area = L × W
- Given: Area of Square - Area of Rectangle = 500 cm²
- Thus, 1600 - L × W = 500
- L × W = 1600 - 500 = 1100 cm²
Step 3: Solving for Length and Width
- We have two equations:
1. L + W = 80
2. L × W = 1100
Using the first equation to express W in terms of L:
- W = 80 - L
Substituting into the second equation:
- L(80 - L) = 1100
- 80L - L² = 1100
- L² - 80L + 1100 = 0
Step 4: Finding Roots
Now, we can apply the quadratic formula:
- L = [80 ± √(80² - 4 × 1 × 1100)] / (2 × 1)
- L = [80 ± √(6400 - 4400)] / 2
- L = [80 ± √2000] / 2
- L = [80 ± 20√5] / 2
- L = 40 ± 10√5
Thus, we can find W:
- W = 80 - L
- The area of the rectangle, L × W, is confirmed to be 1100 cm².
Conclusion
The area of the rectangle is indeed 1100 cm², which corresponds to option 'C'.
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