The most of manufacturing x units of a commodity is `27+15x+3x^(2)`. Find the output where AC = MC.

Answer

To solve the problem where we need to find the output \( x \) such that the Average Cost (AC) equals the Marginal Cost (MC), we can follow these steps:

### Step 1: Define the Cost Function
The total cost function for manufacturing \( x \) units of a commodity is given by:
\[ C(x) = 27 + 15x + 3x^2 \]

### Step 2: Calculate the Average Cost (AC)
The Average Cost (AC) is defined as the total cost divided by the number of units produced:
\[ AC = \frac{C(x)}{x} = \frac{27 + 15x + 3x^2}{x} \]
Simplifying this:
\[ AC = \frac{27}{x} + 15 + 3x \]

### Step 3: Calculate the Marginal Cost (MC)
The Marginal Cost (MC) is the derivative of the total cost function with respect to \( x \):
\[ MC = \frac{dC}{dx} = \frac{d}{dx}(27 + 15x + 3x^2) \]
Calculating the derivative:
\[ MC = 0 + 15 + 6x = 15 + 6x \]

### Step 4: Set AC equal to MC
To find the output where AC equals MC, we set the two equations equal to each other:
\[ \frac{27}{x} + 15 + 3x = 15 + 6x \]

### Step 5: Simplify the Equation
First, we can cancel \( 15 \) from both sides:
\[ \frac{27}{x} + 3x = 6x \]
Now, rearranging gives:
\[ \frac{27}{x} = 6x - 3x \]
\[ \frac{27}{x} = 3x \]

### Step 6: Multiply through by \( x \) to eliminate the fraction
Multiplying both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 27 = 3x^2 \]

### Step 7: Solve for \( x \)
Dividing both sides by \( 3 \):
\[ x^2 = 9 \]
Taking the square root of both sides gives:
\[ x = 3 \quad \text{(since output cannot be negative)} \]

### Conclusion
The output where Average Cost equals Marginal Cost is:
\[ \boxed{3} \]