Introduction

Finding the side of a square when given its area is a fundamental concept in geometry and can provide a good operational scenario to understand square roots and numerical methods. The area of a square is given by the formula:

[text{Area} text{side}^2]

Given that the area of the square is 2025 square units, we will explore how to find the side length using both direct calculation and the Newton-Raphson method.

Direct Calculation of Side Length

To find the length of the side of a square with an area of 2025 square units:

Start with the formula for the area of a square: Set up the equation: Solve for the side length by taking the square root:

Mathematically, this can be represented as:

[text{side} sqrt{2025} 45 text{ units}]

This straightforward method gives us the side length of 45 units.

Using the Newton-Raphson Method

The Newton-Raphson method is a powerful technique for finding the roots of a function. When applied to finding the square root, it can converge to the solution more efficiently than direct calculation, especially for larger or non-perfect squares. The method involves an iterative process.

Steps for the Newton-Raphson Method

Start with an initial guess for the solution. A good initial guess can significantly reduce the number of iterations required. Update the guess using the following formula: Continue updating the guess until the value stabilizes to the desired precision. Verify the result using the direct calculation method.

Let's go through an example using the area of 2025 square units.

Example Calculation

Given the area as 2025, we can use the Newton-Raphson method to find the side length:

Initial guess: (x_1 32 times 1.4 approx 44.8) Next guess: Continue until convergence. In our specific case, after a few iterations, the side length converges to 45 units.

Comparison with Direct Calculation

Direct calculation using the square root function:

[text{side} sqrt{2025} 45 text{ units}]

Using the Newton-Raphson method:

First guess: (x_1 1) Second guess: (x_2 1013) Converges to 45 units after several iterations. First guess: (x_1 40) Converges to 45 units after fewer iterations.

For a more challenging area like 2030, where an integer solution is not obvious, the Newton-Raphson method remains effective:

First guess: (x_1 45) Converges to 45.00000001 units after a few iterations.

Additional Example

For a different original side length, if we assume a cube that was tripled in size to 12.6515 inches, the original side length would be:

[text{Original side length} frac{12.6515}{3} 4.2172text{ inches}]

This demonstrates the practical application of these methods in real-world scenarios where precise measurements are crucial.

Conclusion

The methods discussed here provide a robust approach to finding the side length of a square given its area. The direct calculation method is straightforward and efficient for perfect squares, while the Newton-Raphson method offers an iterative approach that can handle non-perfect squares and provide high precision with fewer calculations.