
Question content area top

Part 1

A firm manufactures a product that sells for 

​$2525 

per unit. Variable cost per unit is 

$ 7$7 

and fixed cost per period is 

$ 1440.$1440. 

Capacity per period is 

12001200 

units.

​(a) Develop an algebraic statement for the revenue function and the cost function.

​(b) Determine the number of units required to be sold to break even.

​(c) Compute the​ break-even point as a percent of capacity.

​(d) Compute the​ break-even point in sales dollars.

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**.

---

### 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).

A firm manufactures a product that sells for $25 per unit. Variable cost per unit is $7 and fixed cost per period is $1,440. Capacity per period is 1,200 units.

(a) Develop an algebraic statement for the revenue function and the cost function.
(b) Determine the number of units required to be sold to break even.
(c) Compute the break-even point as a percent of capacity.
(d) Compute the break-even point in sales dollars.

Learn how to perform a break-even analysis for a manufacturing firm with a product selling for $25 per unit, variable costs of $7, and fixed costs of $1,440. This step-by-step guide covers revenue and cost functions, break-even point calculations, and percentage of capacity used. Suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation