A firm has the following total cost and demand functions :
`C(x) = (x^(3))/(3) - 10x^(2) + 300 x`, where x stand for output. Calculate output at which the
Average cost is equal to the marginal cost.

Answer

To solve the problem where we need to find the output \( x \) at which the average cost is equal to the marginal cost, we will follow these steps:

### Step 1: Define the Total Cost Function
The total cost function is given as:
\[ C(x) = \frac{x^3}{3} - 10x^2 + 300x \]

### Step 2: Calculate the Average Cost
The average cost (AC) is defined as the total cost divided by the quantity \( x \):
\[ AC(x) = \frac{C(x)}{x} = \frac{\frac{x^3}{3} - 10x^2 + 300x}{x} \]
Simplifying this, we get:
\[ AC(x) = \frac{x^3}{3x} - \frac{10x^2}{x} + \frac{300x}{x} = \frac{x^2}{3} - 10x + 300 \]

### Step 3: Calculate the Marginal Cost
The marginal cost (MC) is the derivative of the total cost function with respect to \( x \):
\[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}\left(\frac{x^3}{3} - 10x^2 + 300x\right) \]
Calculating the derivative:
\[ MC(x) = \frac{1}{3} \cdot 3x^2 - 20x + 300 = x^2 - 20x + 300 \]

### Step 4: Set Average Cost Equal to Marginal Cost
Now we set the average cost equal to the marginal cost:
\[ \frac{x^2}{3} - 10x + 300 = x^2 - 20x + 300 \]

### Step 5: Simplify the Equation
Subtract \( 300 \) from both sides:
\[ \frac{x^2}{3} - 10x = x^2 - 20x \]
Rearranging gives:
\[ \frac{x^2}{3} - x^2 + 10x - 20x = 0 \]
This simplifies to:
\[ \frac{x^2}{3} - \frac{3x^2}{3} + 10x - 20x = 0 \]
\[ -\frac{2x^2}{3} - 10x = 0 \]

### Step 6: Multiply Through by -3 to Eliminate the Fraction
Multiplying the entire equation by -3:
\[ 2x^2 + 30x = 0 \]

### Step 7: Factor the Equation
Factoring out \( 2x \):
\[ 2x(x + 15) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
1. \( 2x = 0 \) → \( x = 0 \)
2. \( x + 15 = 0 \) → \( x = -15 \) (not a valid output since output cannot be negative)

Thus, the only valid solution is:
\[ x = 15 \]

### Conclusion
The output at which the average cost is equal to the marginal cost is:
\[ \boxed{15} \]