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?

---

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.