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The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio
- a)1:4
- b)2:9
- c)1:3
- d)3:8
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The area of a rectangle and the square of its perimeter are in the rat...
Let 'a' and 'b' be the length of sides of the rectangle. (a > b)
Area of the rectangle = a*b
Perimeter of the rectangle = 2*(a+b)
Area of the rectangle = a*b
Perimeter of the rectangle = 2*(a+b)
Most Upvoted Answer
The area of a rectangle and the square of its perimeter are in the rat...
Given:
The area of a rectangle and the square of its perimeter are in the ratio 1:25.
To find:
The ratio of the lengths of the shorter and longer sides of the rectangle.
Let's assume:
Let the length of the shorter side of the rectangle be x.
Let the length of the longer side of the rectangle be y.
Perimeter of the rectangle:
The perimeter of a rectangle is given by the formula:
Perimeter = 2(length + breadth)
Given that the square of the perimeter is in the ratio 1:25, we can write:
(2(x + y))^2 = 25
Expanding the equation:
4(x^2 + 2xy + y^2) = 25
Area of the rectangle:
The area of a rectangle is given by the formula:
Area = length x breadth
Given that the area is in the ratio 1:25, we can write:
x * y = 1
Simplifying the equation:
y = 1/x
Substituting the value of y in the perimeter equation:
4(x^2 + 2x(1/x) + (1/x)^2) = 25
Simplifying the equation:
4(x^2 + 2 + 1/x^2) = 25
4x^2 + 8 + 4/x^2 = 25
Multiplying by x^2 to remove the fraction:
4x^4 + 8x^2 + 4 = 25x^2
Simplifying the equation:
4x^4 + 8x^2 - 25x^2 + 4 = 0
4x^4 - 17x^2 + 4 = 0
Factoring the quadratic equation:
(2x^2 - 1)(2x^2 - 4) = 0
The possible values for x are:
2x^2 - 1 = 0 or 2x^2 - 4 = 0
Solving the equations:
2x^2 = 1 or 2x^2 = 4
x^2 = 1/2 or x^2 = 2
Taking the positive square root:
x = √(1/2) or x = √2
The ratio of the lengths of the shorter and longer sides:
Since y = 1/x, we can conclude that:
y = √2 or y = 1/√(1/2)
Simplifying the equation:
y = √2 or y = √2
Therefore, the lengths of the shorter and longer sides of the rectangle are in the ratio 1:√2, which can be approximated as 1:1.41.
The correct answer is option A) 1:4, which is not obtained from the calculations. Therefore, there might be an error in the given options, or the question might have been incorrectly transcribed.
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