A manufacturer sets up a manufacturing unit involving fixed cost of Rs 1,45,000 and the variable cost for producing each unit is 250 Rs. Each unit can be sold at Rs 750.Determine i) Cost function ii) Average Cost Function iii) Revenue Function iv) Average Revenue Function v)Profit Function.

Let's break down the problem step by step.

### Given:
- **Fixed cost (F):** Rs 1,45,000
- **Variable cost per unit (V):** Rs 250
- **Selling price per unit (S):** Rs 750

### i) Cost Function (C(x)):
The total cost function includes both the fixed costs and the variable costs, which depend on the number of units produced, \( x \).

\[
C(x) = \text{Fixed Cost} + \text{Variable Cost per unit} \times \text{Number of units produced}
\]
\[
C(x) = 1,45,000 + 250x
\]

### ii) Average Cost Function (AC(x)):
The average cost function is the total cost divided by the number of units produced.

\[
AC(x) = \frac{C(x)}{x} = \frac{1,45,000 + 250x}{x}
\]
\[
AC(x) = \frac{1,45,000}{x} + 250
\]

### iii) Revenue Function (R(x)):
The revenue function is the total revenue obtained from selling \( x \) units at the selling price \( S \).

\[
R(x) = \text{Selling Price per unit} \times \text{Number of units sold}
\]
\[
R(x) = 750x
\]

### iv) Average Revenue Function (AR(x)):
The average revenue function is the total revenue divided by the number of units sold.

\[
AR(x) = \frac{R(x)}{x} = \frac{750x}{x}
\]
\[
AR(x) = 750
\]

### v) Profit Function (P(x)):
The profit function is the difference between the total revenue and the total cost.

\[
P(x) = R(x) - C(x)
\]
\[
P(x) = 750x - (1,45,000 + 250x)
\]
\[
P(x) = 750x - 1,45,000 - 250x
\]
\[
P(x) = 500x - 1,45,000
\]

### Summary:
1. **Cost Function:** \( C(x) = 1,45,000 + 250x \)
2. **Average Cost Function:** \( AC(x) = \frac{1,45,000}{x} + 250 \)
3. **Revenue Function:** \( R(x) = 750x \)
4. **Average Revenue Function:** \( AR(x) = 750 \)
5. **Profit Function:** \( P(x) = 500x - 1,45,000 \)

Would you like more details or have any questions?

### Further Questions:
1. At what quantity of production does the company break even?
2. What is the profit when 500 units are sold?
3. How does the profit change if the selling price increases to Rs 800 per unit?
4. What is the total cost when 1,000 units are produced?
5. How does the average cost function behave as the production quantity increases?

### Tip:
When analyzing cost and profit functions, always check the break-even point to understand the minimum production level required to avoid losses.

Explore the mathematical analysis of manufacturing costs, revenue, and profit functions. Learn how to determine cost functions, average cost functions, revenue functions, average revenue functions, and profit functions using given fixed costs, variable costs, and selling prices.

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