Understanding the Cost of Fencing a Rectangular Park: A Real-life Application

In this article, we explore a real-world problem involving the cost of fencing a rectangular park and demonstrate the calculation of its dimensions based on the given cost and ratio. This guide is designed to help readers understand the process of solving such problems using simple algebra and basic arithmetic.

Problem Statement

The cost of fencing a rectangular park is given as Rs 3648, and the cost of fencing per meter is Rs 28.50. Additionally, the length and width of the park are in a specific ratio of 5:3. Our task is to determine the length and width of the park.

Solution Steps

We will start by defining the variables and using the given ratio to express the dimensions in terms of a single variable. Then, we will use the cost of fencing to find the actual dimensions.

Step 1: Establish Variables and Convert the Ratio

Let the length of the park be denoted as ( x ) and the width as ( y ). Given the ratio of length to width is 5:3, we can express this ratio as:

[ frac{x}{y} frac{5}{3} ]

Multiplying both sides by 3y, we get:

[ 3x 5y ]

Solving for ( y ), we find:

[ y frac{3}{5}x ]

Step 2: Calculate the Perimeter of the Park

The perimeter of a rectangle is given by:

[ 2(x y) ]

Substituting ( y frac{3}{5}x ) into the formula, we get:

[ 2left(x frac{3}{5}xright) 2 left( frac{8}{5}x right) frac{16}{5}x ]

Step 3: Use the Cost of Fencing to Find ( x )

The total cost of fencing is the perimeter multiplied by the cost per meter:

[ text{Total Cost} text{Perimeter} times text{Cost per meter} ]

Given the total cost is Rs 3648 and the cost per meter is Rs 28.50, we have:

[ 3648 frac{16}{5}x times 28.50 ]

Dividing both sides by 28.50, we get:

[ frac{3648}{28.50} frac{16}{5}x ]

By simplifying, we find:

[ 128 frac{16}{5}x ]

Multiplying both sides by ( frac{5}{16} ), we obtain:

[ x 40 , text{meters} ]

Using ( y frac{3}{5}x ), we find:

[ y frac{3}{5} times 40 24 , text{meters} ]

Conclusion

The length and width of the rectangular park are 40 meters and 24 meters, respectively.

Verification Through Alternative Method

Another method to solve this problem involves using the total cost and the cost per meter to find the perimeter first. Dividing the total cost by the cost per meter, we get:

[ 3648 / 28.50 128 , text{meters} ]

The perimeter of the rectangle is:

[ 2(Length Width) 128 ]

Substituting the ratio ( Length 5x ) and ( Width 3x ), we have:

[ 2(5x 3x) 128 ]

[ 2(8x) 128 ]

[ 16x 128 ]

[ x 8 , text{meters} ]

Thus, the length and width are:

[ Length 5x 5 times 8 40 , text{meters} ]

[ Width 3x 3 times 8 24 , text{meters} ]

This confirms our earlier solution. The dimensions of the rectangular park are 40 meters long and 24 meters wide, matching the data provided.