Importer imports an item for Rs. 12/- per unit cost and his fixed cost is Rs. 2000/- per day. He sells this item Rs. 18/- per unit cost. Consider x is a number of units import and sells in a day. i) Write down the total cost and total revenue functions. ii) Draw the graphs of cost and iii) Using the graphs, find a) the break-even point b) the loss when 200 units are imports and sell c) the profit when 600 units are imports and sell d) the output sells required to obtain a profit of Rs. 2400/- a day

Let's solve this problem step by step.

### i) Total Cost and Total Revenue Functions

Given:
- Cost price per unit = Rs. 12
- Fixed cost = Rs. 2000 per day
- Selling price per unit = Rs. 18
- Let \( x \) be the number of units imported and sold in a day.

**Total Cost (C(x)):**
- The total cost is the sum of the fixed cost and the variable cost (which is dependent on the number of units \( x \)).
- Variable cost = Cost per unit \(\times\) Number of units = \( 12x \)
- Therefore, the total cost function is:
  \[
  C(x) = 12x + 2000
  \]

**Total Revenue (R(x)):**
- Revenue is the amount of money earned from selling the units.
- Revenue per unit = Rs. 18
- Therefore, the total revenue function is:
  \[
  R(x) = 18x
  \]

### ii) Draw the Graphs of Cost and Revenue

To draw the graphs, we'll plot the two functions \( C(x) = 12x + 2000 \) and \( R(x) = 18x \) on the same axes.

### iii) Find the following using the graphs:

#### a) Break-even Point
The break-even point occurs when Total Revenue equals Total Cost, i.e., \( R(x) = C(x) \).

- Set \( 18x = 12x + 2000 \)
- Simplifying, \( 6x = 2000 \)
- \( x = \frac{2000}{6} \approx 333.33 \)

So, the break-even point is at approximately \( x = 334 \) units. At this point, the importer neither makes a profit nor incurs a loss.

#### b) Loss When 200 Units are Imported and Sold

To find the loss when 200 units are imported and sold, calculate the difference between the total cost and total revenue at \( x = 200 \).

- Total cost at \( x = 200 \):
  \[
  C(200) = 12(200) + 2000 = 2400 + 2000 = 4400 \, \text{Rs.}
  \]
- Total revenue at \( x = 200 \):
  \[
  R(200) = 18(200) = 3600 \, \text{Rs.}
  \]
- Loss = \( C(200) - R(200) = 4400 - 3600 = 800 \, \text{Rs.} \)

So, the loss when 200 units are imported and sold is Rs. 800.

#### c) Profit When 600 Units are Imported and Sold

To find the profit when 600 units are imported and sold, calculate the difference between the total revenue and total cost at \( x = 600 \).

- Total cost at \( x = 600 \):
  \[
  C(600) = 12(600) + 2000 = 7200 + 2000 = 9200 \, \text{Rs.}
  \]
- Total revenue at \( x = 600 \):
  \[
  R(600) = 18(600) = 10800 \, \text{Rs.}
  \]
- Profit = \( R(600) - C(600) = 10800 - 9200 = 1600 \, \text{Rs.} \)

So, the profit when 600 units are imported and sold is Rs. 1600.

#### d) Output Sells Required to Obtain a Profit of Rs. 2400/- a Day

To find the number of units \( x \) required to achieve a profit of Rs. 2400, set up the profit equation and solve for \( x \).

- Profit \( P(x) = R(x) - C(x) \)
- Set \( P(x) = 2400 \), so:
  \[
  2400 = 18x - (12x + 2000)
  \]
  Simplifying:
  \[
  2400 = 6x - 2000
  \]
  \[
  6x = 4400
  \]
  \[
  x = \frac{4400}{6} \approx 733.33
  \]

So, approximately 734 units need to be sold to achieve a profit of Rs. 2400 per day.

Would you like me to plot the graphs or clarify any of these steps?

### Related Questions:
1. How can the break-even point be determined algebraically?
2. What is the significance of the fixed cost in calculating profit and loss?
3. How does the selling price impact the total revenue?
4. What would happen to the break-even point if the fixed cost increased?
5. How would the profit equation change if the selling price per unit was reduced?

**Tip:** Always check the break-even point before scaling production to ensure that operations are profitable at your desired output level.

Explore the detailed analysis of an importer's daily operations, including cost functions, revenue functions, break-even points, and profit and loss calculations. Learn how to determine break-even points, losses, profits, and required sales for specific daily profits. Suitable for advanced high school students and beyond.

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