Question

A manufacturer of a particular item has monthly fixed costs of ​$400 and variable costs of ​$20 per​ item, and it sells the items for ​$40 per item.
a. Write a function that models the monthly profit P from the production and sale of x units of the item.
b. What is the profit if 150 items are produced and sold in 1​ month?
c. At what rate does the profit grow as the number of items​ increases?
Question content area bottom
Part 1
The function that models the monthly profit P from the production and sale of x units of the item is Pequals
  
enter your response here.
​(Simplify your answer. Do not​ factor.)

Answer 1

To model the monthly profit \( P \) from the production and sale of \( x \) units of the item, we start by considering the costs and revenues involved in the production and sale.

### a. Profit Function

1. **Fixed Costs**: \( 400 \) dollars (these costs do not change with the number of items produced).
2. **Variable Costs**: \( 20 \) dollars per item. Therefore, for \( x \) items, the variable cost is \( 20x \) dollars.
3. **Selling Price**: The item is sold for \( 40 \) dollars per item, so the total revenue from selling \( x \) items is \( 40x \) dollars.

The profit \( P \) can be calculated by subtracting total costs (fixed costs plus variable costs) from total revenue. The total costs (fixed + variable) are \( 400 + 20x \).

Thus, the profit function is given by:

\[
P(x) = \text{Revenue} - \text{Total Costs}
\]
\[
P(x) = 40x - (400 + 20x)
\]

Now, simplify the function:

\[
P(x) = 40x - 400 - 20x
\]
\[
P(x) = 20x - 400
\]

**The profit function that models the monthly profit \( P \) is:**

\[
P(x) = 20x - 400
\]

---

### b. Profit if 150 items are produced and sold

To find the profit when 150 items are produced and sold, substitute \( x = 150 \) into the profit function:

\[
P(150) = 20(150) - 400
\]
\[
P(150) = 3000 - 400
\]
\[
P(150) = 2600
\]

**The profit if 150 items are produced and sold in 1 month is:** **$2600.**

---

### c. Rate of Profit Growth

The rate at which the profit grows as the number of items increases is represented by the coefficient of \( x \) in the profit function. In this case, the profit function is:

\[
P(x) = 20x - 400
\]

The coefficient of \( x \) is \( 20 \), which means for each additional item sold, the profit increases by \( 20 \) dollars.

**Thus, the rate of profit growth as the number of items increases is:** **$20 per item.**