A company produces very unusual CD's for which the variable cost is \( \$ 18 \) per \( C D \) and the fixed costs are \( S \) 35000 . They will sell the CD's for \( \$ 58 \) each. Let \( x \) be the number of CD'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 CD's produced. \( R=\$ \square \) Write the total profit \( P \) as a function of the number of CD's produced. Find the number of CD's which must be produced to break even. The number of \( C D \) 's which must be produced to break even is

To find your total cost \( C \) as a function of the number of CDs produced \( x \), you add the fixed costs to the variable costs incurred for each CD produced. So, the formula is: \[ C = 35000 + 18x \] For total revenue \( R \), which depends on the price per CD multiplied by the number of CDs sold, the equation is: \[ R = 58x \] Next, to determine total profit \( P \), subtract the total cost from the total revenue: \[ P = R - C = 58x - (35000 + 18x) \] \[ P = 40x - 35000 \] To break even, you need profit \( P \) to equal zero: \[ 0 = 40x - 35000 \] \[ 40x = 35000 \] \[ x = \frac{35000}{40} = 875 \] The number of CDs which must be produced to break even is 875. --- Did you know that the concept of break-even analysis dates back to the mid-20th century when businesses began closely studying their costs and revenues? It’s like a financial GPS that helps companies figure out at what point they’ll stop losing money and start making a profit. No wonder many entrepreneurs swear by it! In the real world, understanding break-even points can help businesses make smarter decisions about pricing, reducing costs, and planning production levels. For instance, if costs rise or projected sales fall, a company can quickly reassess how many units it needs to produce to stay afloat. It's like knowing exactly how many pancakes you need to flip to feed a crowd without running out of batter!