I. Suppose that a firm manifestine a prost and sells them for $50 eah. The lost of the production and the sales of the product is $200,000 fur each unit produces and sold.
write the profit function for the production and sale of x units. Also, SKetch, the.
profit, revenue ili
and Cost funition.

Let's break this down step by step:

1. **Revenue function**: Revenue is the total income the firm makes from selling products. If the firm sells each unit for $50, and $ x $ represents the number of units sold, the revenue function $ R(x) $ is:

   \[
   R(x) = 50x
   \]
   
   This is because the firm earns $50 for each of the $ x $ units sold.

2. **Cost function**: The cost is the total expense incurred for producing and selling $ x $ units. We are given that the cost for production and sales is $200,000 per unit. So, the cost function $ C(x) $ is:

   \[
   C(x) = 200,000x
   \]

3. **Profit function**: Profit is the difference between the revenue and the cost. Therefore, the profit function $ P(x) $ is:

   \[
   P(x) = R(x) - C(x) = 50x - 200,000x = -199,950x
   \]

   This shows that for each unit sold, the firm is losing $199,950 in profit, meaning the firm operates at a loss.

### Sketching the functions

- **Revenue function** $ R(x) = 50x $: This is a straight line with a positive slope of 50, passing through the origin. The graph will rise as more units are sold.
  
- **Cost function** $ C(x) = 200,000x $: This is also a straight line, but with a much steeper slope of 200,000, passing through the origin. The cost grows much faster than revenue as production increases.

- **Profit function** $ P(x) = -199,950x $: This is a straight line with a negative slope, showing a loss that increases linearly as more units are produced and sold.

I will now provide a sketch of these functions for clarity.

#### Want to see the sketch? 

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### Related Questions:
1. How would the profit function change if the cost per unit decreases?
2. What happens to the break-even point if the revenue per unit increases to $200?
3. How would a fixed cost (e.g., a $50,000 setup cost) impact the cost and profit functions?
4. What if the revenue per unit starts declining after producing a certain number of units?
5. How would the graph look if the cost was a quadratic function due to increasing costs for higher production?

**Tip:** The slope of the profit function indicates how profit changes with each additional unit produced. A negative slope means a loss per unit.

Suppose that a firm manufactures a product and sells them for $50 each. The cost of production and sales of the product is $200,000 for each unit produced and sold. Write the profit function for the production and sale of x units. Also, sketch the profit, revenue, and cost function.

Explore the profit, revenue, and cost functions for a firm that manufactures and sells products. This linear problem involves algebraic concepts, where revenue and cost are calculated per unit sold. Learn how to write and analyze these functions, with a detailed solution and sketch. Suitable for students in Grades 10-12.

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