If the total cost function for a production of x units of a commodity is given by `3/4x^(2)–7x+27`, then the number of units produced for which MC = AC is

Options

A. 4
B. 6
C. 9
D. 36

Answer

To find the number of units produced for which Marginal Cost (MC) equals Average Cost (AC), we will follow these steps:

### Step 1: Define the Total Cost Function
The total cost function is given as:
\[ C(x) = \frac{3}{4}x^2 - 7x + 27 \]

### Step 2: Calculate Marginal Cost (MC)
Marginal Cost is the derivative of the total cost function with respect to \( x \):
\[ MC = \frac{dC}{dx} \]

Differentiating \( C(x) \):
\[ MC = \frac{d}{dx} \left( \frac{3}{4}x^2 - 7x + 27 \right) \]
Using the power rule:
\[ MC = \frac{3}{2}x - 7 \]

### Step 3: Calculate Average Cost (AC)
Average Cost is defined as the total cost divided by the number of units produced:
\[ AC = \frac{C(x)}{x} = \frac{\frac{3}{4}x^2 - 7x + 27}{x} \]
Simplifying:
\[ AC = \frac{3}{4}x - 7 + \frac{27}{x} \]

### Step 4: Set MC equal to AC
Now, we set the Marginal Cost equal to Average Cost:
\[ \frac{3}{2}x - 7 = \frac{3}{4}x - 7 + \frac{27}{x} \]

### Step 5: Solve for \( x \)
First, simplify the equation:
\[ \frac{3}{2}x - 7 + 7 = \frac{3}{4}x + \frac{27}{x} \]
This simplifies to:
\[ \frac{3}{2}x = \frac{3}{4}x + \frac{27}{x} \]

Now, multiply through by \( 4x \) to eliminate the fraction:
\[ 4x \cdot \frac{3}{2}x = 4x \cdot \frac{3}{4}x + 4x \cdot \frac{27}{x} \]
This gives:
\[ 6x^2 = 3x^2 + 108 \]

Now, rearranging gives:
\[ 6x^2 - 3x^2 - 108 = 0 \]
\[ 3x^2 - 108 = 0 \]
\[ 3x^2 = 108 \]
\[ x^2 = 36 \]
\[ x = 6 \quad (\text{since } x \text{ must be positive}) \]

### Conclusion
The number of units produced for which Marginal Cost equals Average Cost is:
\[ \boxed{6} \]