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We are given the revenue function $ R(x) = 15x $ and the cost function $ C(x) = 10x + 110 $, where $ x $ represents the number of units produced. The goal is to determine the profit function and calculate how many units need to be produced for a profit of $575.

### Step 1: Profit Function
The profit function $ P(x) $ is calculated as the difference between the revenue and the cost:
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions:
\[
P(x) = 15x - (10x + 110)
\]
Simplify the expression:
\[
P(x) = 15x - 10x - 110 = 5x - 110
\]

### Step 2: Solve for $ x $ when the profit is $575
We want to find $ x $ such that $ P(x) = 575 $. Set up the equation:
\[
5x - 110 = 575
\]
Add 110 to both sides:
\[
5x = 685
\]
Now, divide by 5:
\[
x = \frac{685}{5} = 137
\]

### Conclusion
To achieve a profit of $575, 137 units must be produced.

Would you like more details on any step or have any further questions?

Here are five related questions to consider:
1. What would be the break-even point where profit is zero?
2. How would changes in the cost function affect the profit margin?
3. What happens if the revenue per unit decreases?
4. How would fixed costs (like the 110 in $ C(x) $) influence production decisions?
5. How many units would need to be produced for a profit of $1000?

**Tip:** In business problems, understanding how fixed costs and variable costs affect the profit function is essential for maximizing profitability.

The revenue for a business, as a function of units produced, x, is shown below by R(x). C(x) represents the cost of producing x units. Calculate the profit function and then determine how many units must be produced for the business to make a profit of $575. R(x) = 15x, C(x) = 10x + 110

This math problem involves calculating the profit function for a business using the revenue function R(x) = 15x and cost function C(x) = 10x + 110. Learn how to determine how many units must be produced to achieve a profit of $575 with step-by-step guidance. Suitable for high school students in Grades 9-12.

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