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Let's analyze and solve the questions in the image you provided.

### Given functions:
- **Cost function**: \( C(x) = 15x + 3000 \)
- **Revenue function**: \( R(x) = 45x + 1500 \)
  where \( x \) represents the number of items produced and sold.

Now let's solve each part:

### i) How many items must be produced so that the cost is less than 4500?

We need to find \( x \) such that the cost \( C(x) < 4500 \).

\[
C(x) = 15x + 3000
\]
Setting the inequality:
\[
15x + 3000 < 4500
\]
Solving for \( x \):
\[
15x < 4500 - 3000
\]
\[
15x < 1500
\]
\[
x < \frac{1500}{15} = 100
\]
Thus, the number of items must be **less than 100** for the cost to be less than 4500.

### ii) What will be the minimum quantities of items to be sold so that the revenue is greater than or equal to 10,500 rupees?

We need to solve for \( x \) such that \( R(x) \geq 10500 \).

\[
R(x) = 45x + 1500
\]
Setting the inequality:
\[
45x + 1500 \geq 10500
\]
Solving for \( x \):
\[
45x \geq 10500 - 1500
\]
\[
45x \geq 9000
\]
\[
x \geq \frac{9000}{45} = 200
\]
Thus, the minimum number of items to be sold for revenue to be greater than or equal to 10500 is **200** items.

### iii) If the number of items sold lies between 300 to 400, then what will be the revenue?

We are asked to find the revenue when \( x \) is between 300 and 400, i.e., for \( 300 \leq x \leq 400 \).

Using the revenue function \( R(x) = 45x + 1500 \):

For \( x = 300 \):
\[
R(300) = 45(300) + 1500 = 13500 + 1500 = 15000
\]

For \( x = 400 \):
\[
R(400) = 45(400) + 1500 = 18000 + 1500 = 19500
\]

So, the revenue will be between **15000 and 19500** when the number of items sold lies between 300 and 400.

### OR) How many items must be sold to gain some profit?

To make a profit, the revenue must exceed the cost, i.e., \( R(x) > C(x) \).

Setting the inequality:
\[
45x + 1500 > 15x + 3000
\]
Solving for \( x \):
\[
45x - 15x > 3000 - 1500
\]
\[
30x > 1500
\]
\[
x > \frac{1500}{30} = 50
\]
Thus, the company must sell more than **50 items** to gain some profit.

---

Would you like more details on any part of this? Here are some related questions for deeper understanding:

1. How would the results change if the cost function had a different fixed cost?
2. What would happen to the minimum quantity required for profit if the selling price per item was lowered?
3. How would you compute the break-even point for this cost and revenue system?
4. What is the profit if 400 items are sold?
5. How do changes in fixed cost or variable cost affect the profit margin?

**Tip**: When solving inequalities, always check if the solution needs to be rounded or exact, especially in real-world scenarios like production or sales.

A company produces certain items. The manager maintains a record of cost and revenue of these items on a daily basis. The cost and revenue functions of the product are given by C(x) = 15x + 3000 and R(x) = 45x + 1500 respectively, where x is the number of items produced and sold. Answer the following: i) How many items must be produced so that the cost is less than 4500? ii) What will be the minimum quantities of items to be sold so that the revenue is greater than or equal to 10500 rupees? iii) If the number of items sold lies between 300 to 400, what will be the revenue? OR) How many items must be sold to gain some profit?

Learn how to solve cost and revenue-related problems using linear functions. This case study involves analyzing a company's cost and revenue functions, solving for quantities to minimize costs, maximize revenue, and determine the break-even point for profit.

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