If the total cost of producing x units of a commodity is given by `C(x)=(1)/(3)x^(2)+x^(2)-15x+300`, then the marginal cost when x=5 is

Options

A. Rs 25
B. Rs 20
C. Rs 30
D. Rs 50

Answer

To find the marginal cost when \( x = 5 \), we need to follow these steps:

### Step 1: Write down the total cost function.
The total cost function is given as:
\[ C(x) = \frac{1}{3}x^3 + x^2 - 15x + 300 \]

### Step 2: Differentiate the total cost function to find the marginal cost.
The marginal cost (MC) is the derivative of the total cost function with respect to \( x \):
\[ MC = \frac{dC}{dx} \]

### Step 3: Differentiate the function.
Using the power rule of differentiation:
\[ \frac{d}{dx}\left(\frac{1}{3}x^3\right) = x^2, \quad \frac{d}{dx}(x^2) = 2x, \quad \frac{d}{dx}(-15x) = -15, \quad \frac{d}{dx}(300) = 0 \]
Thus, we have:
\[ MC = x^2 + 2x - 15 \]

### Step 4: Substitute \( x = 5 \) into the marginal cost function.
Now, we substitute \( x = 5 \) into the marginal cost equation:
\[ MC = (5)^2 + 2(5) - 15 \]

### Step 5: Calculate the values.
Calculating each term:
\[ (5)^2 = 25, \quad 2(5) = 10 \]
So,
\[ MC = 25 + 10 - 15 \]
\[ MC = 35 - 15 = 20 \]

### Final Answer:
The marginal cost when \( x = 5 \) is \( \text{Rs. } 20 \).

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