A company spends as much as x2 - 370x +63000 pesos to manufacture x number of bags in a week. If the production cost budget is at most 30,000 pesos a week, how many bags should the company manufacture in a week without overspending? 

The problem involves finding how many bags the company can manufacture in a week without exceeding their budget of 30,000 pesos. The cost function for producing \( x \) bags is given by:

\[
C(x) = x^2 - 370x + 63000
\]

We are told that the production cost must be **at most** 30,000 pesos, so we can set up the inequality:

\[
x^2 - 370x + 63000 \leq 30000
\]

### Step 1: Simplify the inequality
First, subtract 30,000 from both sides of the inequality to move all terms to one side:

\[
x^2 - 370x + 63000 - 30000 \leq 0
\]

This simplifies to:

\[
x^2 - 370x + 33000 \leq 0
\]

### Step 2: Solve the quadratic inequality
Now, solve the quadratic equation:

\[
x^2 - 370x + 33000 = 0
\]

We can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -370 \), and \( c = 33000 \).

\[
x = \frac{-(-370) \pm \sqrt{(-370)^2 - 4(1)(33000)}}{2(1)}
\]

\[
x = \frac{370 \pm \sqrt{136900 - 132000}}{2}
\]

\[
x = \frac{370 \pm \sqrt{4900}}{2}
\]

\[
x = \frac{370 \pm 70}{2}
\]

### Step 3: Find the roots
Now, calculate the two possible solutions for \( x \):

\[
x = \frac{370 + 70}{2} = \frac{440}{2} = 220
\]
\[
x = \frac{370 - 70}{2} = \frac{300}{2} = 150
\]

### Step 4: Interpret the solution
The quadratic inequality \( x^2 - 370x + 33000 \leq 0 \) is satisfied between the two roots, \( x = 150 \) and \( x = 220 \). Therefore, the company can manufacture between **150 and 220 bags** per week without exceeding the production budget of 30,000 pesos.

### Final Answer:
The company should manufacture between **150 and 220 bags** per week to stay within the budget.

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

Here are 5 related questions to explore further:
1. How would the solution change if the production cost budget increased to 40,000 pesos?
2. Can you derive the production cost if 180 bags are manufactured?
3. What is the minimum production cost for manufacturing 150 bags?
4. How can you graph the quadratic function to visualize the solution?
5. What happens to the cost function if fixed costs are introduced?

**Tip:** When solving quadratic inequalities, always check the boundary points to ensure they satisfy the inequality.

A company spends as much as x^2 - 370x + 63000 pesos to manufacture x number of bags in a week. If the production cost budget is at most 30,000 pesos a week, how many bags should the company manufacture in a week without overspending?

Discover how to solve the quadratic inequality x^2 - 370x + 63000 ≤ 30,000 to determine how many bags a company can manufacture within a production budget. This problem covers key algebraic concepts such as quadratic equations and inequalities, making it suitable for students in Grades 9-11.

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