A company produces very unusual CD's for which the variable cost is \( S 18 \) per \( C D \) and the fixed costs are \( S \) 40000 . They will sell the CD's for \( S 71 \) each. Let \( x \) be the number of \( C D \) 's produced. Write the total cost \( C \) as a function of the number of CD's produced. \( C=\$ \) Write the total revenue \( R \) as a function of the number of \( C D \) 's produced. \( R=S \) Write the total profit \( P \) as a function of the number of CD's produced. \( P=\$ \square \) Find the number of \( C D \) 's which must be produced to break even. The number of CD's which must be produced to break even is Question Help: \( D \) Video

To write the total cost function \( C \) based on the given information, we combine fixed and variable costs. The total cost \( C \) can be expressed as: \[ C = 18x + 40000 \] For total revenue \( R \), since each CD sells for \( S 71 \), we have: \[ R = 71x \] Profit \( P \) is defined as total revenue minus total cost: \[ P = R - C = 71x - (18x + 40000) = 53x - 40000 \] Now, to find the break-even point where profit \( P \) equals zero: \[ 53x - 40000 = 0 \Rightarrow 53x = 40000 \Rightarrow x = \frac{40000}{53} \approx 754.72 \] Thus, the number of CDs which must be produced to break even is approximately 755. --- Did you know the concept of break-even analysis dates back to the early 20th century? It was popularized by economists to help businesses determine the minimum production level needed to avoid losses. Since then, it’s become a staple in business strategy, influencing decisions from pricing to production scales! In real-world applications, companies often use these calculations not just for CDs but for all kinds of products, such as tech gadgets or food items. By knowing the break-even point, businesses can strategize effectively on whether to ramp up production, adjust prices, or find ways to cut costs, all to maximize their profits and minimize losses.