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The total-cost function, denoted by \(C(x)\), represents the total cost associated with producing \(x\) units of a product. It typically includes fixed costs (costs that remain constant regardless of the production level) and variable costs (costs that vary with the production level). In the given exercise, the total-cost function is specified as \(C(x) = 24x + 50,000\). Here,

• 24x represents the variable cost, where 24 is the cost per unit of production, and
• 50,000 is the fixed cost that does not change with the production level.

Incorporating both fixed and variable costs is crucial for accurately understanding the expenses involved in the production process.

The total-revenue function, denoted by \(R(x)\), represents the total revenue generated from selling \(x\) units of a product. Revenue is typically calculated as the price per unit times the number of units sold. In the given exercise, the total-revenue function is \(R(x) = 40x\). Here,

• 40 represents the sale price per unit, and
• x is the number of units sold.

Understanding the total-revenue function is essential because it helps businesses determine their earnings from sales, which is a crucial aspect of profitability analysis.

The break-even point is the production level at which total revenue equals total cost, resulting in zero profit. It is a critical milestone for any business, as it determines the minimum quantity of units that must be sold to cover all costs. To find the break-even point, we need the total-profit function, denoted by \(P(x)\), which is the difference between the total-revenue function and the total-cost function: \[P(x) = R(x) - C(x)\]From the given functions,\[P(x) = 40x - (24x + 50,000)\]. Simplifying this, we get:\[P(x) = 16x - 50,000\]. Setting the total-profit function equal to zero to find the break-even point:\[16x - 50,000 = 0\]. Solving for \(x\), we get \[x = \frac{50,000}{16} = 3,125\].This means the business needs to produce and sell 3,125 units to break even. Understanding the break-even point helps businesses plan production efficiently and make informed financial decisions.