Let's go through the problem step by step.

Given:

• Cost to produce each unit: $140
• Selling price per unit: $230
• Fixed cost: $1,200

a) Break-even point:

The break-even point occurs when total costs equal total revenue.

1. Revenue function $R(x)$:
   $R(x) = 230x$, where $x$ is the number of units sold.

2. Cost function $C(x)$:
   $C(x) = 140x + 1200$, where $140x$ is the variable cost and $1,200 is the fixed cost.

3. Break-even point condition:
   $R(x) = C(x)$

$230x = 140x + 1200$ Solving for $x$:

$230x - 140x = 1200$ $90x = 1200$ $x = \frac{1200}{90} = 13.33$

So, the break-even point is at approximately 14 units (since the number of units must be a whole number).

b) Profit function:

Profit is the difference between revenue and cost.

$P(x) = R(x) - C(x)$ Substitute the functions:

$P(x) = 230x - (140x + 1200)$ Simplify:

$P(x) = 230x - 140x - 1200$ $P(x) = 90x - 1200$

Thus, the profit function is $P(x) = 90x - 1200$.

c) Profit for 2,000 units:

Substitute $x = 2000$ into the profit function:

$P(2000) = 90(2000) - 1200$ $P(2000) = 180,000 - 1200$ $P(2000) = 178,800$

So, the profit for producing and selling 2,000 units is $178,800.

Would you like further details or clarifications? Here are 5 questions to expand on this:

1. How does the profit change with the increase in the cost per unit?
2. What if the fixed cost increased to $2,000? How would the break-even point be affected?
3. Can the profit function be used to predict losses? If so, how?
4. What happens if the selling price is reduced by 10%? How does that affect the break-even point?
5. How does the break-even analysis help businesses in pricing their products?

Tip: Break-even analysis is a crucial tool in understanding the relationship between cost, revenue, and profit. It helps businesses make informed decisions on production levels and pricing strategies.