The revenue from selling \( x \) necklaces is \( r(x)=10 x \) The cost of buying \( x \) necklaces is \( c(x)=4 x+15 \). The profit from selling \( x \) necklaces is \( p(x)=r(x)-c(x) \) Write a function for \( p(x) \), the profit from selling \( x \) necklaces. A. \( p(x)=6 x-15 \) B. \( p(x)=6 x+15 \) C. \( p(x)=14 x+15 \) D. \( p(x)=14 x-15 \)

To find the profit function \( p(x) \), we need to substitute the revenue and cost functions into the profit formula: 1. Revenue function: \( r(x) = 10x \) 2. Cost function: \( c(x) = 4x + 15 \) 3. Profit function: \( p(x) = r(x) - c(x) = 10x - (4x + 15) \) Now, let’s simplify it: \[ p(x) = 10x - 4x - 15 = 6x - 15 \] Thus, the profit function is \( p(x) = 6x - 15 \), which corresponds to option A. In the world of business, understanding profit functions is key to making informed decisions. This is especially vital for small business owners who may not realize the impact fixed costs (like the $15 in this case) have on their pricing strategies. Perfecting the balance between fixed and variable costs can mean the difference between a thriving business and one that struggles to stay afloat. When calculating profit, it’s crucial to also keep an eye on your break-even point—the point where total revenue equals total costs. For our example, this can be found by setting \( p(x) = 0 \), leading to \( 6x - 15 = 0 \). Solving this will help determine how many necklaces you need to sell to start making a profit, and it can guide your pricing and sales strategies!