The Newton-Raphson method is given by the iteration formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Step 1: Define the function and its derivative

Given equation:

$f(x) = x^3 + 4x^2 - 10$

Compute the derivative:

$f'(x) = 3x^2 + 8x$

Step 2: Compute the first iteration

Initial guess:
$x_0 = 1.5$

Evaluate $f(x_0)$:

$f(1.5) = (1.5)^3 + 4(1.5)^2 - 10$

$= 3.375 + 9 - 10 = 2.375$

Evaluate $f'(x_0)$:

$f'(1.5) = 3(1.5)^2 + 8(1.5)$

$= 3(2.25) + 12 = 6.75 + 12 = 18.75$

Compute the next approximation:

$x_1 = 1.5 - \frac{2.375}{18.75}$

$x_1 = 1.5 - 0.12666667$

$x_1 = 1.37333333$

Step 3: Check Accuracy

Since $x_1 = 1.37333333$ is accurate to at least three significant figures (1.37), we stop here.

Step 4: Choose the correct answer

The correct answer is:
$\boxed{1.373333333} \quad (\text{Option a})$

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Related Questions:

1. What is the Newton-Raphson method used for in real-world applications?
2. How do we determine if Newton-Raphson will converge to a root?
3. What are the advantages of Newton-Raphson compared to other root-finding methods?
4. What happens if the derivative $f'(x)$ is close to zero?
5. How can Newton-Raphson be modified for multiple roots?

Tip: Always check if the derivative is close to zero before applying Newton-Raphson, as it can lead to large errors or divergence.