A manufacturing company finds that the daily cost of producing x items of a

product is given by C(x) = 210x + 7000.

(i) If each item is sold for Rs. 350, find the minimum number that must be

produced and sold

daily to ensure no loss.

(ii) If the selling price is increased by Rs. 35 per piece, what would be the break-

even point.

Let's solve the problem step by step.

### (i) Minimum Number of Items to Ensure No Loss

Given:
- Cost function: \( C(x) = 210x + 7000 \)
- Selling price per item: Rs. 350

To ensure no loss, the revenue from selling \( x \) items should be at least equal to the cost of producing \( x \) items.

Revenue \( R(x) \) is given by:
\[
R(x) = 350x
\]

For no loss (break-even point):
\[
R(x) = C(x)
\]
\[
350x = 210x + 7000
\]

Now, solve for \( x \):
\[
350x - 210x = 7000
\]
\[
140x = 7000
\]
\[
x = \frac{7000}{140} = 50
\]

So, the minimum number of items that must be produced and sold daily to ensure no loss is **50**.

### (ii) Break-even Point with Increased Selling Price

New selling price per item: Rs. \( 350 + 35 = 385 \)

New revenue function \( R(x) \) is given by:
\[
R(x) = 385x
\]

At the break-even point:
\[
R(x) = C(x)
\]
\[
385x = 210x + 7000
\]

Now, solve for \( x \):
\[
385x - 210x = 7000
\]
\[
175x = 7000
\]
\[
x = \frac{7000}{175} = 40
\]

So, the break-even point with the increased selling price is **40** items.

---

### Summary:
1. **Minimum number of items to avoid loss:** 50 items.
2. **Break-even point with increased price:** 40 items.

---

Would you like more details or have any questions?

Here are some related questions for further practice:
1. What would be the profit if 60 items are produced and sold at Rs. 350 per item?
2. How does the cost function change if the fixed cost increases by Rs. 1000?
3. Calculate the profit margin if 70 items are sold at Rs. 385 each.
4. Determine the loss if only 30 items are produced and sold at Rs. 350 each.
5. How would the break-even point change if the variable cost per item decreases by Rs. 20?

**Tip:** Always ensure to understand the difference between fixed costs and variable costs when analyzing cost functions.

A manufacturing company finds that the daily cost of producing x items of a product is given by C(x) = 210x + 7000. (i) If each item is sold for Rs. 350, find the minimum number that must be produced and sold daily to ensure no loss. (ii) If the selling price is increased by Rs. 35 per piece, what would be the break-even point.

Learn how to find the minimum number of items a manufacturing company must produce and sell to avoid loss using algebraic equations. This problem covers break-even analysis, linear equations, and key formulas for calculating cost and revenue. Suitable for Grades 9-12.

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