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
- Fixed Costs: \( 400 \) dollars (these costs do not change with the number of items produced).
- Variable Costs: \( 20 \) dollars per item. Therefore, for \( x \) items, the variable cost is \( 20x \) dollars.
- 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.