Understanding the Problem: Calculating Levelling Cost

This problem involves calculating the cost of levelling a rectangular park based on information derived from a square field. We are given the area of the square field, and the dimensions of the rectangular park are related to the side length of the square field. Finally, we need to find the total cost of levelling the park at a given rate per square meter.

Step 1: Find the Side Length of the Square Field

The area of a square is calculated by squaring its side length. We are given the area of the square field is 1764 m\({}^2\). Let the side length of the square field be \(s\) meters.

The formula for the area of a square is:

\(\text{Area} = s^2\)

Given the area is 1764 m\({}^2\), we have:

\(s^2 = 1764\)

To find the side \(s\), we need to calculate the square root of 1764:

\(s = \sqrt{1764}\)

We can find the square root. Let's try estimating: \(40^2 = 1600\), \(50^2 = 2500\). The number ends in 4, so the unit digit of its square root must be 2 or 8. Let's try 42:

\(42 \times 42 = 1764\)

So, the side length of the square field is 42 meters.

Step 2: Determine the Dimensions of the Rectangular Park

We are told that the breadth of the rectangular park is \(\frac{1}{6}\)th of the side of the square field.

Breadth of rectangular park \(b\) = \(\frac{1}{6} \times \text{Side of square field}\)

\(b = \frac{1}{6} \times 42\) meters

\(b = 7\) meters

We are also told that the length of the rectangular park is four times its breadth.

Length of rectangular park \(l\) = \(4 \times \text{Breadth of rectangular park}\)

\(l = 4 \times 7\) meters

\(l = 28\) meters

So, the dimensions of the rectangular park are length = 28 meters and breadth = 7 meters.

Step 3: Calculate the Area of the Rectangular Park

The area of a rectangle is calculated by multiplying its length and breadth.

\(\text{Area of rectangle} = \text{Length} \times \text{Breadth}\)

\(\text{Area of rectangular park} = 28 \times 7\) m\({}^2\)

\(\text{Area of rectangular park} = 196\) m\({}^2\)

Step 4: Calculate the Cost of Levelling the Park

The cost of levelling the park is given as Rs. 30 per square meter. To find the total cost, we multiply the area of the park by the cost per square meter.

\(\text{Total cost} = \text{Area of rectangular park} \times \text{Cost per } \text{m}^2\)

\(\text{Total cost} = 196 \times 30\)

\(\text{Total cost} = 5880\)

The total cost of levelling the rectangular park is Rs. 5880.

Summary of Calculations

The calculated cost of levelling the park is Rs. 5880.

Revision Table: Area and Perimeter Formulas

Additional Information: Working with Square Roots

Finding the square root of a number is the inverse operation of squaring a number. For example, since \(42^2 = 1764\), the square root of 1764 is 42. When dealing with areas, the side length must be a positive value, so we take the positive square root.

To find the square root of a larger number manually, you can use methods like prime factorization or the long division method for square roots. For 1764, prime factorization is \(1764 = 2 \times 882 = 2 \times 2 \times 441 = 2^2 \times 21^2 = 2^2 \times (3 \times 7)^2 = 2^2 \times 3^2 \times 7^2\). The square root is \(\sqrt{2^2 \times 3^2 \times 7^2} = 2 \times 3 \times 7 = 6 \times 7 = 42\).