A small company manufactures a certain item and sells it online. The company has a business model where the cost C, in dollars, to make x items is given by the equation C=203x+50 and the revenue R, in dollars, made by selling x items is given by the equation R=10x. The break-even point is the point where the cost and tevenue equations intersect. Be sure to answer both Part A and Part B
Answer
Break-Even Point Analysis: Understanding Cost and Revenue Equations
When running a small business, it’s critical to understand your costs and revenue to determine when your company begins to make a profit. This key moment in business economics is known as the break-even point. It’s the point where your revenue equals your total cost — meaning you are not making a loss or profit yet, but just breaking even.
📘 What Are Cost and Revenue Equations?
For this analysis, the business has two important equations:
Where x represents the number of items produced or sold.
✅ Revenue depends only on how many items are sold, earning $10 per item.
🧮 Part A: Finding the Break-Even Point
The break-even point occurs where cost equals revenue. So we need to set:
Substitute the given equations into the formula:
Step 1: Rearranging the Equation
We move all terms involving x to one side and constants to the other:
Step 2: Solving for x
Now divide both sides of the equation by 193:
This tells us something very important about the business model: the cost per item is greater than the revenue per item. That means for every item produced, the company is actually losing money.
🔍 Interpreting the Result
Let’s take a closer look. The company is spending $203 per item to manufacture, but only earning $10 per item. No matter how many items they produce, the cost will always be much greater than the revenue.
➤ The more items produced, the greater the loss.
📈 Part B: Graphical Representation of Cost vs. Revenue
Let’s visualize this using the equations:
- Cost Function: Starts at $50 and increases steeply at $203 per item
- Revenue Function: Starts at $0 and increases slowly at $10 per item
On a graph, the cost line will always stay above the revenue line, which confirms that they never intersect at any positive value of x.
Sample Points:
| x (Items) | Cost (C = 203x + 50) | Revenue (R = 10x) | Profit/Loss |
|---|---|---|---|
| 0 | $50 | $0 | –$50 (Loss) |
| 1 | $253 | $10 | –$243 (Loss) |
| 2 | $456 | $20 | –$436 (Loss) |
| 5 | $1,065 | $50 | –$1,015 (Loss) |
| 10 | $2,080 | $100 | –$1,980 (Loss) |
💡 Business Insight
This kind of scenario is a red flag for any business. Selling a product for less than it costs to produce is unsustainable. Here are some key takeaways:
- Production Cost = $203/item
- Sales Revenue = $10/item
- Net Loss per item = $193
- Fixed Cost = $50
📊 How to Improve the Business Model
To achieve profitability, the business can consider:
- Increase the Selling Price: Raise the unit price above $203 to cover production and make a profit.
- Reduce Production Cost: Look for cheaper materials, automate production, or negotiate bulk deals.
- Reduce Fixed Costs: Eliminate unnecessary overhead or operational expenses.
- Reevaluate the Product: If customers won’t pay over $203, consider whether the product has market fit.
📚 Final Thoughts
Break-even analysis is crucial for small businesses to determine when they become financially viable. In this scenario, the cost structure is misaligned with the revenue model, meaning the company will face continuous losses unless changes are made. Understanding the relationship between cost, revenue, and production volume helps entrepreneurs make smart, data-driven decisions.