a manufracturer estimates that it cost$14 to produce each unit of a particular commodity thats cells for $23 per unit.There is also a fixed cost of dollar 1200.

The manufacturer incurs two types of costs: **fixed costs** and **variable costs**.

- **Fixed cost**: \$1200 (This cost does not change with the number of units produced.)
- **Variable cost per unit**: \$14 (This is the cost to produce each additional unit.)

The **selling price** of the commodity is \$23 per unit. Now, let's break down the problem.

### Total Cost (C):
The total cost of production consists of the fixed cost and the variable cost. If $x$ represents the number of units produced, then the total cost $C(x)$ is given by:
\[
C(x) = 1200 + 14x
\]

### Total Revenue (R):
The revenue is calculated by multiplying the number of units sold by the selling price per unit. So, the total revenue $R(x)$ is:
\[
R(x) = 23x
\]

### Profit (P):
The profit is the difference between the total revenue and the total cost. So, the profit $P(x)$ is:
\[
P(x) = R(x) - C(x)
\]
Substituting the formulas for revenue and cost:
\[
P(x) = 23x - (1200 + 14x) = 23x - 1200 - 14x = 9x - 1200
\]

The manufacturer makes a profit of \$9 per unit after covering the variable costs, but they need to overcome the fixed cost of \$1200 before making a total profit.

### Break-even point:
The break-even point occurs when profit equals zero, i.e., when $P(x) = 0$. To find the break-even number of units:
\[
9x - 1200 = 0
\]
\[
9x = 1200
\]
\[
x = \frac{1200}{9} \approx 133.33
\]

So, the manufacturer needs to sell approximately 134 units to break even (since they can't sell a fraction of a unit).

Would you like further details on any part of this? Here are some related questions:

1. How do you calculate the profit for different unit sales?
2. What happens if the fixed cost changes?
3. How do changes in the selling price affect the break-even point?
4. What is the impact of producing more units on the overall profit?
5. How can you minimize the cost per unit to increase profitability?

**Tip**: Understanding the break-even point helps businesses know how many units they need to sell to avoid a loss!

A manufacturer estimates that it costs $14 to produce each unit of a particular commodity that sells for $23 per unit. There is also a fixed cost of $1200.

Learn how to calculate the break-even point for a manufacturer who incurs a fixed cost of $1200 and a variable cost of $14 per unit, while selling each unit for $23. This detailed solution includes step-by-step breakdowns of total cost, revenue, profit, and the break-even point. Ideal for students in Grades 8-10.

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