Question

For a textile industry, the revenue of selling \( x \) tons of yarn is \( R(x) = 4x \) and the cost of producing \( x \) tons of yarn is \( C(x) = 10 + 2x + 3x^{2/3} \). The industry decided to calculate the break-even (when revenue is equal to the cost) for \( x \) tons of yarn using the Newton-Raphson method. Assuming the initial break-even of 8 tons, the break-even (in tons) after the first iteration is:

• A. 6
• B. 8
• C. 12
• D. 14

Hint:

In the Newton-Raphson method, the function and its derivative need to be calculated iteratively until the desired accuracy is reached.

Answer 1

The Correct Option is D

Step 1: Define the break-even condition.
The break-even point occurs when \( R(x) = C(x) \). That is: \[ 4x = 10 + 2x + 3x^{2/3} \] Step 2: Define the function \( f(x) \) for the break-even condition.
We define the function \( f(x) = R(x) - C(x) \), so: \[ f(x) = 4x - (10 + 2x + 3x^{2/3}) \] Thus, we get: \[ f(x) = 2x - 10 - 3x^{2/3} \] Step 3: Derivative of \( f(x) \).
Now, differentiate \( f(x) \) to get \( f'(x) \): \[ f'(x) = 2 - 2x^{-1/3} \] Step 4: Apply the Newton-Raphson method.
Using the Newton-Raphson iteration formula: \[ x_{{new}} = x_{{old}} - \frac{f(x_{{old}})}{f'(x_{{old}})} \] Starting with \( x_{{old}} = 8 \), we calculate: \[ f(8) = 2(8) - 10 - 3(8^{2/3}) = 16 - 10 - 3(4) = 16 - 10 - 12 = -6 \] \[ f'(8) = 2 - 2(8^{-1/3}) = 2 - 2 \times \frac{1}{2} = 1 \] Thus, the first iteration gives: \[ x_{{new}} = 8 - \frac{-6}{1} = 8 + 6 = 14 \] Step 5: Conclusion.
Thus, the break-even point after the first iteration is \( \boxed{14} \).