Let's break down the problem step by step to answer each part.

(a) Find the production cost (in dollars) per item

We are given the following information:

• The total cost to produce 25 items is $\$8,950$.
• The total cost to produce 55 items is $\$10,150$.

We can use these two data points to determine the production cost per item by finding the slope of the cost function.

Step 1: Calculate the slope (variable cost per item)

The slope represents the variable cost per item. The slope $m$ between two points $(x_1, C_1)$ and $(x_2, C_2)$ on the cost function is given by:

$m = \frac{C_2 - C_1}{x_2 - x_1}$

Substituting the given values:

$m = \frac{10,150 - 8,950}{55 - 25} = \frac{1,200}{30} = 40$

So, the production cost per item is $\boxed{40}$ dollars.

(b) Find the company's linear profit function

To find the profit function $P(x)$, we need to consider both the revenue function and the cost function.

Step 1: Find the revenue function

We know the company has a revenue of $\$9,000$ from selling 75 items. The revenue per item is:

$\text{Revenue per item} = \frac{9,000}{75} = 120$

So, the revenue function $R(x)$ is:

$R(x) = 120x$

Step 2: Determine the cost function

We already know the variable cost per item is $\$40$. However, we also need to determine the fixed costs.

Given that the cost function is linear and has the form $C(x) = 40x + b$, we can use one of the given points to solve for $b$ (the fixed costs).

Using the point $(25, 8,950)$:

$8,950 = 40(25) + b$ $8,950 = 1,000 + b$ $b = 8,950 - 1,000 = 7,950$

So, the cost function $C(x)$ is:

$C(x) = 40x + 7,950$

Step 3: Find the profit function

The profit function $P(x)$ is given by the difference between the revenue and the cost functions:

$P(x) = R(x) - C(x)$

Substituting the revenue and cost functions:

$P(x) = 120x - (40x + 7,950)$ $P(x) = 120x - 40x - 7,950$ $P(x) = 80x - 7,950$

So, the company's linear profit function is:

$\boxed{P(x) = 80x - 7,950}$

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Would you like to explore these details further or ask any questions? Here are five related questions you might find helpful:

1. What would the profit be if the company produced and sold 100 items?
2. How would the profit function change if the cost per item increased by $5?
3. If the company wanted to break even, how many items would they need to sell?
4. How would the fixed costs affect the profit if they were reduced by $2,000?
5. What would happen to the profit function if the selling price per item dropped to $110?

Tip: The slope of the profit function indicates how much profit increases for each additional item sold.