https://file.aiquickdraw.com/mathbot/file/1726087564196jdgmelhm.jpg

The problem from the image states:

- Revenue function: \( R(x) = -0.025x^2 + 18x \)
- Cost function: \( C(x) = 5x + 19 \)

Where \( x \) is the number of trucker hats produced and sold. The task is to **find the production level \( x \)** that maximizes the **weekly profit**.

### Step 1: Profit function
The profit \( P(x) \) is the difference between the revenue \( R(x) \) and the cost \( C(x) \):
\[
P(x) = R(x) - C(x)
\]
Substitute the given functions:
\[
P(x) = \left(-0.025x^2 + 18x\right) - \left(5x + 19\right)
\]
Simplify:
\[
P(x) = -0.025x^2 + 18x - 5x - 19
\]
\[
P(x) = -0.025x^2 + 13x - 19
\]

### Step 2: Find the maximum profit
Since this is a quadratic function of the form \( P(x) = ax^2 + bx + c \), the maximum profit occurs at the vertex of the parabola. The formula for the \( x \)-coordinate of the vertex is:
\[
x = \frac{-b}{2a}
\]
Here, \( a = -0.025 \) and \( b = 13 \), so:
\[
x = \frac{-13}{2(-0.025)} = \frac{-13}{-0.05} = 260
\]

Thus, the production level \( x = 260 \) hats maximizes the weekly profit.

Would you like more details or clarifications?

### Relative Questions:
1. What is the total revenue at the optimal production level?
2. What is the total cost when producing 260 hats?
3. What is the profit at this optimal production level?
4. How does changing the coefficient of \( x^2 \) affect the production level that maximizes profit?
5. Can you describe the economic meaning of the coefficients in the revenue and cost functions?

### Tip:
The vertex formula \( x = \frac{-b}{2a} \) is useful for finding maximum or minimum values of quadratic functions. Keep it handy for optimization problems!

The weekly revenue from the sale of x trucker hats is R(x) = -0.025x^2 + 18x dollars, and the weekly cost of producing x trucker hats is C(x) = 5x + 19 dollars. Find the production level x that maximizes the weekly profit.

This problem focuses on finding the production level of trucker hats that maximizes weekly profit. The revenue function is R(x) = -0.025x^2 + 18x, and the cost function is C(x) = 5x + 19. Learn how to calculate the optimal number of hats using quadratic equations and profit maximization techniques.

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