Solving for the Length of a Rectangle Given Perimeter and Ratio

This article explores the method to determine the length of a rectangle when the length and breadth are given in a specific ratio and the perimeter of the rectangle is known. We will use algebraic methods to solve the problem and provide a step-by-step guide to understanding the solution.

Problem Statement

The ratio of the length to the breadth of a rectangle is 5:2. The respective ratio of its perimeter and area is 1:3. What is the length of the rectangle?

Solution

Let the length of the rectangle be 5x and the breadth be 2x, where x is a positive constant.

Step 1: Calculate the Perimeter and Area

First, we need to express the perimeter and area of the rectangle in terms of x.

Perimeter (P):

P 2(length breadth) 2(5x 2x) 2(7x) 14x

Area (A):

A length × breadth 5x × 2x 12

Step 2: Set Up the Ratio of Perimeter to Area

According to the problem, the ratio of the perimeter to the area is 1:3. Therefore, we can write:

(frac{P}{A} frac{1}{3})

Substituting the expressions for P and A, we get:

(frac{14x}{1^2} frac{1}{3})

Step 3: Cross-Multiply to Solve for x

Cross-multiplying, we get:

14x × 3 12

42x 12

Now, let's rearrange the equation:

12 - 42x 0

2x(5x - 21) 0

This gives us two solutions:

x 0 (not a valid solution as it would make the dimensions zero) 5x - 21 0

Solving 5x - 21 0:

5x 21

x (frac{21}{5}) 4.2

Step 4: Calculate the Length

The length of the rectangle is:

5x 5 × 4.2 21 units

Conclusion

The length of the rectangle is 21 units.

Additional Examples and Problem-Solving Processes

Let's consider a second scenario to solidify our understanding.

Additional Example 1

Given that the length and width of a rectangle are in the ratio 5:2, and the perimeter is 70 cm, find the length of the rectangle.

Given: (frac{L}{W} frac{5}{2}) (Thus, (W frac{2L}{5})) We know: Perimeter (P 2(L W) 2L 2W) Substitute (W frac{2L}{5}) into the formula: 70 2L 2(left(frac{2L}{5}right)) Solve the equation: 70 2L (frac{4L}{5}) 70 (frac{10L}{5} frac{4L}{5}) 70 (frac{14L}{5}) Distribute the 70: (frac{70 times 5}{14} L) 25 L or L 25 cm

Additional Example 2

A rectangle length L and width W are in a ratio 5:2, and the perimeter is 70 cm. Length L 5W/2.

Given:

Length L (frac{5W}{2}) Perimeter 2L 2W 70 cm

Substitution:

70 2((frac{5W}{2})) 2W

70 5W 2W 7W

W 10 cm

L (frac{5 times 10}{2}) 25 cm

The length of the rectangle is 25 cm.

Conclusion

This article has provided a thorough walkthrough of how to solve for the length and width of a rectangle given a ratio and perimeter. By following these steps and examples, one can confidently approach similar geometric problems.