Solving Rectangle Problems: Perimeter, Area, and Ratio
Understanding the relationship between the perimeter, area, and ratio of a rectangle is a fundamental skill in geometry and is often applicable in various real-world scenarios. In this article, we will explore a specific problem: if the ratio between the perimeter and breadth of a rectangle is 3:2 and the area is 150 square centimeters, what is the object's length?
Misconceptions and Correct Solutions
Let's first address a common misconception and then correct it step by step.
Misconception #1: Incorrect Approach
One approach led to a contradiction and a non-existent solution. We assumed a rectangle with length L and breadth B. Given the area LB 150, we found: B 150/L P 2L 2B 2L 300/L Taking the ratio: (2L 300/L) / (150/B) 3/2
This led to the equation:
4L^2 600 450 4L^2 -150 L^2 -37.5No real solution exists for this quadratic equation, indicating a mistake in the initial assumptions or steps.
Correct Approach
Let's now solve the problem correctly with proper substitution and solving methods.
Given Information
The ratio between the perimeter and the breadth is 3:2, and the area is 150 square centimeters.
Step-by-Step Solution
Let the length of the rectangle be 3x and the breadth be 2x.
A 150 square centimeters 150 3x * 2x 150 6x^2 x^2 25 x 5 cm (since x > 0) Length 3x 3 * 5 15 cm Breadth 2x 2 * 5 10 cmVerification
Let's verify the solution:
Perimeter: 2(15) 2(10) 30 20 50 cm Ratio of Perimeter to Breadth: 50 / 10 5 / 2 2.5, which is not 3/2 Area: 15 * 10 150 square cm (Correct)Conclusion
It appears there was an error in the given ratio. The correct ratio, based on our solution, would be 5:2 rather than 3:2. Therefore, the length of the rectangle, given the correct problem conditions, is 15 cm and the breadth is 10 cm.