A company finds the total cost of producing 25 items is $8,950, while they can produce 55 items for a total cost of $10,150. They have a revenue of $9,000 by selling 75 items. Answer each of the following. Enter all answers below using exact numbers, and put equations in slope-intercept form. Let *x* be the number of items.

(a)

Find the production cost (in dollars) per item.

(b)

Find the company's linear profit function.

*P*(*x*) = 
 
 

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:

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

---

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.

Learn how to calculate production costs per item and derive linear profit functions using given cost and revenue data. Solve specific numerical examples involving linear equations in cost and profit analysis. Suitable for high school students studying linear equations and business mathematics.

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