Let's break down each part of the problem step by step:

Given Information:

• Selling Price per Unit (Revenue per Unit) = $25
• Variable Cost per Unit = $7
• Fixed Costs per Period = $1,440
• Capacity per Period = 1,200 units

Part (a): Develop the Revenue Function and the Cost Function

1. Revenue Function:

   The revenue function $R(x)$ represents the total revenue generated from selling $x$ units. Given that each unit sells for $25, the revenue function is:

   $R(x) = 25x$ where $x$ is the number of units sold.

2. Cost Function:

   The total cost function $C(x)$ consists of fixed costs and variable costs. The fixed cost is $1,440 per period, and the variable cost is $7 per unit. Thus, the cost function is:

   $C(x) = 1440 + 7x$ where $x$ is the number of units produced and sold.

Part (b): Determine the Number of Units Required to Break Even

The break-even point occurs when total revenue equals total costs. This can be determined by solving the equation $R(x) = C(x)$.

$25x = 1440 + 7x$

Simplifying this equation:

$25x - 7x = 1440$ $18x = 1440$ $x = \frac{1440}{18} = 80$

So, the firm needs to sell 80 units to break even.

Part (c): Compute the Break-Even Point as a Percent of Capacity

The capacity is 1,200 units. To find the break-even point as a percentage of capacity:

$\text{Break-even percent} = \left(\frac{80}{1200}\right) \times 100 = 6.67\%$

Thus, the break-even point is 6.67% of the firm's capacity.

Part (d): Compute the Break-Even Point in Sales Dollars

The break-even point in sales dollars is the revenue generated at the break-even point, which can be calculated by multiplying the number of units at the break-even point by the price per unit:

$\text{Break-even sales} = 80 \times 25 = 2000$

So, the break-even point in sales dollars is $2,000.

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Summary of Results:

• (a) Revenue function: $R(x) = 25x$, Cost function: $C(x) = 1440 + 7x$
• (b) Break-even units: 80 units
• (c) Break-even point as a percent of capacity: 6.67%
• (d) Break-even point in sales dollars: $2,000

Would you like any further details or explanations?

Related Questions:

1. What happens if the selling price per unit increases to $30?
2. How would a change in fixed costs to $1,600 affect the break-even point?
3. What if the variable cost increases to $10 per unit? How does it change the break-even point?
4. Can you compute the profit if the firm sells 150 units in a period?
5. How would a reduction in capacity to 1,000 units affect the break-even percentage?

Tip:

To minimize the break-even point, firms can either increase the selling price or reduce costs (fixed or variable).