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Analysis Of Profit Function Based On Item Sales

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    Analysis of Profit Function Based on Item Sales Annual profit in thousands of dollars is given by the function, P(x) = 100*sqrt(x – 5) + 3000, where x is the number of items sold, x ≥ 5. 1. Describe the meaning of the number 5 in the formula. Answer: The number 5 in the formula P(x)=100x−5+3000P(x) = 100 \ sqrt{x - 5} + 3000 represents the minimum number of items that must be sold for the profit function to be valid. Explanation: • The formula involves the square root function, x−5 \ sqrt{x - 5}, and for the square root to be real and valid, x−5x - 5 must be non - negative. This means x≥5x \ geq 5 to ensure that the expression inside the square root is non - negative and the profit function can be computed. • At x=5x = 5, the expression x−5 \ sqrt{x - 5} becomes 5−5=0=0 \ sqrt{5 - 5} = \ sqrt{0} = 0, meaning that when 5 items are sold, the profit is 30003000 thousand dollars (because the additional profit term, 100x−5100 \ sqrt{x - 5}, is 0). Conclusion: The number 5 in the formula sets the threshold for the number of items sold, indicating that sales must be at least 5 for the profit function to apply and start generating a profit above the base amount of 3000 thousand dollars. It essentially represents the starting point for calculating the profit. 2. Describe the meaning of the number 3000 in the formula. Use a complete sentence or sentences. Answer: The number 3000 in the formula P(x)=100x−5+3000P(x) = 100 \ sqrt{x - 5} + 3000 represents the base profit in thousands of dollars that the company earns regardless of the number of items sold. This is the fixed amount of profit the company earns when no additional items are sold, corresponding to the situation where x=5x = 5. It can be seen as the starting or guaranteed profit, even before any sales generate additional profit based on the number of items sold. 3. Find the profit for 6 different values of x (show your work). PLEASE use x = 5, 69, 105, 230, 366, and 489 and make a table using Microsoft Excel. Answer: To find the profit for each value of xx, you can follow these calculations and create a table in Excel:

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