Annual profit in thousands of dollars is given by the function, P(x) = 200√ x - 3  + 10,000, where x is the number of items sold in thousands, x ≥ 3. 

describe the meaning of the number 3 in the formula
describe the meaning of the number 10,000 in the formula
find the profit for 5 different values of x
graph the profit function over its given domain; use the 5 values calculated in part 3 to construct the graph and connect these points with a smooth curve in Excel or another graphing utility. Insert the graph in a Word file and attach the graph in a Word file to the class DB thread.
will this profit function have a maximum, if so, what is it?
what steps should the company take to prepare for your answer to part 5?
Post your final draft as a response to this post; use the small group area for collaboration.

1. The number 3 in the formula represents the minimum number of items (in thousands) that need to be sold in order for the profit function to be valid. This means that the function is only defined for values of x that are greater than or equal to 3. If the company sells fewer than 3,000 items, the profit function won't provide a meaningful result.

2. The number 10,000 in the formula represents the fixed amount of profit in thousands of dollars, regardless of the number of items sold. It acts as a constant term added to the variable part of the profit function. This means that even if the company doesn't sell any items (x = 0), the profit would still be $10,000. It can be understood as a baseline profit that exists independently of the number of items sold.

3. To find the profit for different values of x, we simply need to substitute those values into the given profit function P(x). Let's calculate the profit for five different values of x:
a) For x = 3, P(3) = 200√(3) - 3 + 10,000
b) For x = 4, P(4) = 200√(4) - 3 + 10,000
c) For x = 5, P(5) = 200√(5) - 3 + 10,000
d) For x = 6, P(6) = 200√(6) - 3 + 10,000
e) For x = 7, P(7) = 200√(7) - 3 + 10,000

4. To graph the profit function over its given domain, we can use a graphing utility like Microsoft Excel. We will plot the values of x and their corresponding profits, using the five values calculated in step 3. By connecting these points with a smooth curve, we can visualize the profit function. The graph can be inserted into a Word file and attached to the class discussion board thread.

5. The profit function described by the equation P(x) = 200√x - 3 + 10,000 will have a maximum profit. To find the maximum profit, we need to analyze the concavity of the function. If the function is concave down, it will have a maximum. To determine this, we can calculate the second derivative of the profit function and check its sign. If the second derivative is negative, it indicates concavity down, and there will exist a maximum profit. The specific value of the maximum can be found by analyzing critical points within the given domain.

6. To prepare for an analysis of the maximum profit, the company should ensure that the profit function aligns with their business model and assumptions. They should also gather relevant data on the number of items sold and their corresponding profits, ensuring that the range of data falls within the domain of the function. By having accurate and comprehensive data along with a clear understanding of the profit function, the company can make informed decisions based on the maximum profit point.