The Oliver Company plans to market a new product. Based on its market studies, Oliver estimates that it can sell up to 5,500 units in 2005. The selling price will be $5 per unit. Variable costs are estimated to be 50% of total revenue. Fixed costs are estimated to be $5,700 for 2005. How many units should the company sell to break even?

To find the break-even point for the Oliver Company, we need to determine the number of units it should sell so that the total revenue equals the total costs (both variable and fixed costs).

### Step 1: Define the Revenue and Cost Functions

1. **Revenue Function (R):**  
   The revenue generated by selling $ x $ units is:
   \[
   R(x) = 5x
   \]
   where $ 5 $ is the selling price per unit.

2. **Variable Cost Function (VC):**  
   Variable costs are estimated to be 50% of total revenue. So, the variable cost function is:
   \[
   VC(x) = 0.5 \times R(x) = 0.5 \times 5x = 2.5x
   \]

3. **Fixed Costs (FC):**  
   The fixed costs are constant and given as $5,700 for the year 2005.  
   \[
   FC = 5,700
   \]

4. **Total Cost Function (TC):**  
   The total cost is the sum of variable and fixed costs:
   \[
   TC(x) = VC(x) + FC = 2.5x + 5,700
   \]

### Step 2: Set Up the Break-Even Equation

The break-even point occurs when total revenue equals total cost:  
\[
R(x) = TC(x)
\]

Substitute the expressions for $ R(x) $ and $ TC(x) $:
\[
5x = 2.5x + 5,700
\]

### Step 3: Solve for the Break-Even Point

Rearrange the equation to isolate $ x $:
\[
5x - 2.5x = 5,700
\]
\[
2.5x = 5,700
\]
\[
x = \frac{5,700}{2.5}
\]
\[
x = 2,280
\]

### Conclusion

The company must sell **2,280 units** to break even.

### Would you like more details on any part of the solution, or do you have any other questions?

### Related Questions:

1. What will be the profit if the company sells 3,000 units?
2. How much revenue will the company generate at the break-even point?
3. What is the total cost if the company sells 4,000 units?
4. How would an increase in fixed costs to $7,000 affect the break-even point?
5. What is the impact on the break-even point if the variable costs increase to 60% of the total revenue?

**Tip:** Knowing the break-even point helps in understanding how much needs to be sold to cover all costs, which is essential for planning production and sales strategies.

Learn how to calculate the break-even point for the Oliver Company's new product. Understand the concepts of revenue, variable costs, and fixed costs. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation