To determine how many espresso machines need to be sold for the firm to break even, we can use the following formula:

$\text{Break-even point} = \frac{\text{Fixed cost}}{\text{Selling price per machine} - \text{Variable cost per machine}}$

Where:

• Fixed cost = $1,063,201
• Selling price per machine = $388
• Variable cost per machine = $98

Now, substitute these values into the formula:

$\text{Break-even point} = \frac{1,063,201}{388 - 98}$

$\text{Break-even point} = \frac{1,063,201}{290}$

Calculating:

$\text{Break-even point} \approx 3666.21$

Since the number of machines must be a whole number, round to the nearest whole number:

$\text{Break-even point} \approx 3,666$

Thus, the firm needs to sell approximately 3,666 espresso machines to break even.

Would you like further details or have any questions? Here are some related questions to expand on this concept:

1. What happens to the break-even point if the variable cost increases?
2. How does a change in the selling price affect the break-even point?
3. What is the total revenue at the break-even point?
4. How much profit does the firm make after selling 1,000 machines beyond the break-even point?
5. How would a reduction in fixed costs impact the break-even point?

Tip: Understanding the relationship between fixed costs, variable costs, and selling prices is crucial for effective pricing and production strategies in business.