A text-book publisher finds that the production costs directly attribute to each book are of Rs 20 and the fixed costs are Rs 10,000. If each book can be sold for Rs 30, determine the break-even point.

Answer

To determine the break-even point for the textbook publisher, we will follow these steps:

### Step 1: Define the Cost Function
The total cost (C) consists of fixed costs and variable costs. The variable cost per book is Rs 20, and the fixed costs are Rs 10,000.

\[ C(x) = \text{Variable Cost} + \text{Fixed Cost} = 20x + 10,000 \]

### Step 2: Define the Revenue Function
The revenue (R) generated from selling x books, where each book is sold for Rs 30, is given by:

\[ R(x) = \text{Selling Price per Book} \times \text{Number of Books Sold} = 30x \]

### Step 3: Set Up the Profit Function
The profit (P) is defined as the revenue minus the total cost. Therefore, we can express the profit function as:

\[ P(x) = R(x) - C(x) = 30x - (20x + 10,000) \]

### Step 4: Simplify the Profit Function
Now, simplify the profit function:

\[ P(x) = 30x - 20x - 10,000 = 10x - 10,000 \]

### Step 5: Find the Break-even Point
The break-even point occurs when the profit is zero (P(x) = 0). Therefore, we set the profit function equal to zero:

\[ 10x - 10,000 = 0 \]

### Step 6: Solve for x
Now, solve for x:

\[ 10x = 10,000 \]
\[ x = \frac{10,000}{10} = 1,000 \]

Thus, the break-even point is when 1,000 books are sold.

### Final Answer
The break-even point is 1,000 books.

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