A company produces widgets that sell for $15 each. The cost to produce \( x \) widgets is given by \( C(x) = 5x + 200 \). How many widgets must be sold to break even?
["Breaking Even at $15 per Widget: How Many Must You Sell?", "Understanding when a business crosses its break-even point is crucial for sustainable growth. In this example, a company produces widgets priced at $15 each, with a fixed production cost structure defined by the cost function ( C(x) = 5x + 200 ), where ( x ) represents the number of widgets produced. But how many widgets must be sold to break even—where revenue matches total cost?", "### What Is the Break-Even Point?", "The break-even point occurs when total revenue equals total cost. In mathematical terms:", "[\n\ ext{Revenue} = \ ext{Cost}\n]", "Revenue from selling ( x ) widgets at $15 each is:", "[\nR(x) = 15x\n]", "Total cost is given as:", "[\nC(x) = 5x + 200\n]", "### Setting Up the Equation", "To find the break-even quantity, solve:", "[\n15x = 5x + 200\n]", "### Solving for ( x )", "Subtract ( 5x ) from both sides:", "[\n10x = 200\n]", "Now divide both sides by 10:", "[\nx = 20\n]", "### Interpreting the Result", "This means the company must sell 20 widgets to cover all production and overhead costs. At this volume, revenue of ( 20 \ imes 15 = 300 ) dollars equals the total cost of ( 5 \ imes 20 + 200 = 100 + 200 = 300 ) dollars—achieving break-even.", "Below 20 widgets, the company incurs a loss; above 20, profits begin to accumulate.", "### Conclusion", "To ensure financial stability, this widget-producing company needs to sell 20 units per month—the break-even threshold determined by its pricing and cost structure. Monitoring sales volume relative to this point is essential for forecasting profitability and guiding operational decisions.", "---", "Keywords: break even point, widget production cost, revenue vs cost graph, break even calculation, business profitability, manufacturer cost analysis.\nMeta Description: Discover how to calculate the break-even point for widgets selling at $15 each, with a cost function of ( C(x) = 5x + 200 ). Learn why selling 20 units is critical for profitability."]
