It has been determined that the cost of producing x units of a certain item is 8x+2608x+260.  The demand function is given by p=D(x)=56−0.4xp=D(x)=56−0.4x.

Find the profit function.

To find the profit function \( P(x) \), we need to subtract the cost function \( C(x) \) from the revenue function \( R(x) \).

### Step 1: Identify the given functions
- **Cost function**: \( C(x) = 8x + 260 \)
- **Demand function (price per unit)**: \( p = D(x) = 56 - 0.4x \)

### Step 2: Find the revenue function
The revenue function \( R(x) \) is given by:
\[
R(x) = p \times x = (56 - 0.4x) \times x
\]
\[
R(x) = 56x - 0.4x^2
\]

### Step 3: Determine the profit function
The profit function \( P(x) \) is calculated as:
\[
P(x) = R(x) - C(x)
\]
Substituting the values of \( R(x) \) and \( C(x) \):
\[
P(x) = (56x - 0.4x^2) - (8x + 260)
\]

### Step 4: Simplify the expression
Simplifying the expression for \( P(x) \):
\[
P(x) = 56x - 0.4x^2 - 8x - 260
\]
\[
P(x) = (56x - 8x) - 0.4x^2 - 260
\]
\[
P(x) = 48x - 0.4x^2 - 260
\]

So, the profit function is:
\[
P(x) = -0.4x^2 + 48x - 260
\]

Would you like to explore more details or have any further questions?

Here are 8 related questions you might consider:

1. How can you find the value of \( x \) that maximizes the profit?
2. What is the significance of the coefficients in the profit function?
3. How do you determine the break-even points where profit is zero?
4. What is the marginal profit function, and how do you calculate it?
5. How does the cost function affect the overall profit?
6. Can you find the maximum profit by using calculus?
7. What happens to the profit function if fixed costs increase?
8. How does the demand function impact the profit function?

**Tip:** When dealing with quadratic functions like the profit function, the vertex form can help identify maximum or minimum values efficiently.

Learn how to derive the profit function from the given cost and demand functions. This problem involves understanding cost, demand, revenue, and profit functions in economics and algebra. Suitable for students in Grades 10-12.

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