If C(x)=`0.05x - 0.2x^(2) - 5` find the value of output for which the average cost becomes equal to the marginal cost.

Answer

To solve the problem of finding the value of output \( x \) for which the average cost equals the marginal cost, we will follow these steps:

### Step 1: Define the Cost Function
The cost function is given as:
\[ C(x) = 0.05x - 0.2x^2 - 5 \]

### Step 2: Calculate the Average Cost
The average cost \( AC \) is calculated by dividing the total cost \( C(x) \) by the output \( x \):
\[ AC = \frac{C(x)}{x} = \frac{0.05x - 0.2x^2 - 5}{x} \]
This simplifies to:
\[ AC = 0.05 - 0.2x - \frac{5}{x} \]

### Step 3: Calculate the Marginal Cost
The marginal cost \( MC \) is the derivative of the cost function \( C(x) \) with respect to \( x \):
\[ MC = \frac{dC}{dx} = \frac{d}{dx}(0.05x - 0.2x^2 - 5) \]
Using the power rule of differentiation:
\[ MC = 0.05 - 0.4x \]

### Step 4: Set Average Cost Equal to Marginal Cost
To find the output \( x \) where average cost equals marginal cost, we set \( AC = MC \):
\[ 0.05 - 0.2x - \frac{5}{x} = 0.05 - 0.4x \]

### Step 5: Simplify the Equation
Cancelling \( 0.05 \) from both sides, we get:
\[ -0.2x - \frac{5}{x} = -0.4x \]
Rearranging gives:
\[ 0.4x - 0.2x = \frac{5}{x} \]
This simplifies to:
\[ 0.2x = \frac{5}{x} \]

### Step 6: Multiply Both Sides by \( x \)
To eliminate the fraction, multiply both sides by \( x \):
\[ 0.2x^2 = 5 \]

### Step 7: Solve for \( x \)
Dividing both sides by \( 0.2 \):
\[ x^2 = \frac{5}{0.2} = 25 \]
Taking the square root of both sides gives:
\[ x = 5 \]

### Conclusion
The value of output \( x \) for which the average cost equals the marginal cost is:
\[ \boxed{5} \]

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