The manufacturing cost of an item consists of rupes 100 as overheads , material cost rupes 2 per item and the labour cost `( x^(2))/(90) ` for x items , produced . Find the average cost.

Answer

To find the average cost of manufacturing an item given the costs, we can break the problem down into a series of steps:

### Step 1: Identify the components of the total cost
The total cost \( C \) of producing \( x \) items consists of three components:
1. Overheads: \( 100 \) rupees
2. Material cost: \( 2 \) rupees per item, which totals to \( 2x \) for \( x \) items.
3. Labour cost: \( \frac{x^2}{90} \) rupees for \( x \) items.

Thus, the total cost \( C \) can be expressed as:
\[ C = 100 + 2x + \frac{x^2}{90} \]

### Step 2: Calculate the average cost
The average cost \( AC \) is defined as the total cost divided by the number of items produced \( x \):
\[ AC = \frac{C}{x} = \frac{100 + 2x + \frac{x^2}{90}}{x} \]

This simplifies to:
\[ AC = \frac{100}{x} + 2 + \frac{x}{90} \]

### Step 3: Differentiate the average cost with respect to \( x \)
To find the value of \( x \) that minimizes the average cost, we differentiate \( AC \) with respect to \( x \):
\[ \frac{d(AC)}{dx} = -\frac{100}{x^2} + \frac{1}{90} \]

### Step 4: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ -\frac{100}{x^2} + \frac{1}{90} = 0 \]

### Step 5: Solve for \( x^2 \)
Rearranging the equation gives:
\[ \frac{1}{90} = \frac{100}{x^2} \]
Multiplying both sides by \( 90x^2 \) results in:
\[ x^2 = 9000 \]

### Step 6: Calculate \( x \)
Taking the square root of both sides, we find:
\[ x = 30\sqrt{10} \]

### Step 7: Verify if it is a minimum
To confirm that this value of \( x \) minimizes the average cost, we can check the second derivative:
\[ \frac{d^2(AC)}{dx^2} = \frac{200}{x^3} \]
Since this is positive for \( x > 0 \), it indicates that the average cost is minimized at \( x = 30\sqrt{10} \).

### Summary
The average cost of manufacturing the item is minimized when \( x = 30\sqrt{10} \).

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