Using Newton-Raphson method, find the cube root of 24 ?A. 2.884B. 3.256C. 5.231D. 4.526

To find the cube root of 24 using the Newton-Raphson method, follow these steps:

Step 1: Define the function
We want to find the cube root of 24, so we can write the problem as solving the equation:

f(x) = x³ - 24 = 0

The derivative of f(x) is:

f'(x) = 3x²

Step 2: Newton-Raphson iteration formula
The Newton-Raphson iteration formula is:

xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ)

Substituting f(x) and f'(x), the formula becomes:

xₙ₊₁ = xₙ - (xₙ³ - 24) / (3xₙ²)

Step 3: Initial guess
Choose an initial guess close to the expected root. The cube root of 24 is approximately between 2.8 and 3 (since 2.8³ = 21.95 and 3³ = 27). Let’s start with x₀ = 3.

Step 4: Perform iterations
First iteration (n = 0):
x₁ = 3 - (3³ - 24) / (3 × 3²)
= 3 - (27 - 24) / (27)
= 3 - 3 / 27
= 3 - 0.1111
≈ 2.8889

Second iteration (n = 1):
x₂ = 2.8889 - (2.8889³ - 24) / (3 × 2.8889²)
= 2.8889 - (24.1221 - 24) / (3 × 8.3541)
= 2.8889 - 0.1221 / 25.0623
= 2.8889 - 0.0049
≈ 2.8840

Third iteration (n = 2):
x₃ = 2.8840 - (2.8840³ - 24) / (3 × 2.8840²)
= 2.8840 - (23.9981 - 24) / (3 × 8.3161)
= 2.8840 - (-0.0019) / 24.9483
= 2.8840 + 0.00008
≈ 2.8840

Step 5: Conclusion
The iterations converge, and the cube root of 24 is approximately 2.884.

Correct answer: A. 2.884