A company finds the total cost of producing 25 items is $8,750, while they can produce 55 items for a total cost of $10,850. 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.Find the company's linear profit function.

To find the company's linear profit function, we need to determine both the cost function $ C(x) $ and the revenue function $ R(x) $, and then subtract the cost function from the revenue function to get the profit function $ P(x) $.

### Step 1: Find the Cost Function $ C(x) $
The cost function $ C(x) $ is typically linear and can be expressed in the form:
\[
C(x) = mx + b
\]
where $ m $ is the marginal cost (the cost per additional item), and $ b $ is the fixed cost.

We are given two points for the cost function:
- $ C(25) = 8750 $
- $ C(55) = 10850 $

These points can be used to find the slope $ m $:

\[
m = \frac{C(55) - C(25)}{55 - 25} = \frac{10850 - 8750}{30} = \frac{2100}{30} = 70
\]

So, the cost per item is $70. Now, we can substitute one of the points back into the equation to find $ b $:

Using $ C(25) = 8750 $:
\[
8750 = 70(25) + b
\]
\[
8750 = 1750 + b
\]
\[
b = 8750 - 1750 = 7000
\]

Thus, the cost function is:
\[
C(x) = 70x + 7000
\]

### Step 2: Find the Revenue Function $ R(x) $
The revenue function $ R(x) $ is also linear and can be expressed as:
\[
R(x) = px
\]
where $ p $ is the price per item.

We are given the revenue for 75 items:
- $ R(75) = 9000 $

This gives us:
\[
9000 = 75p
\]
\[
p = \frac{9000}{75} = 120
\]

So, the revenue function is:
\[
R(x) = 120x
\]

### Step 3: Find the Profit Function $ P(x) $
The profit function is the difference between the revenue function and the cost function:
\[
P(x) = R(x) - C(x)
\]
Substituting the expressions we found:
\[
P(x) = 120x - (70x + 7000)
\]
Simplifying:
\[
P(x) = 120x - 70x - 7000
\]
\[
P(x) = 50x - 7000
\]

Thus, the company's linear profit function is:
\[
\boxed{P(x) = 50x - 7000}
\]

Would you like any additional details or have any questions?

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**Related Questions:**

1. What is the break-even point where profit equals zero?
2. How many items must be sold to achieve a profit of $5,000?
3. What is the total cost to produce 100 items?
4. If the revenue per item decreases to $100, how does the profit function change?
5. What is the fixed cost if only 10 items are produced?

**Tip:** When analyzing profit, always ensure to calculate both fixed and variable costs separately for accurate insights.

Learn how to calculate a company's linear profit function using cost, revenue, and profit analysis. Solve for the cost function, revenue function, and derive the profit function in a step-by-step manner. Includes calculations based on given data points such as cost per item, total revenue, and fixed costs.

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