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

We are given the company's cost function for manufacturing \( x \) bags per week as:

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

Additionally, we know the company's production cost budget is at most 30,000 pesos per week. This gives us the inequality:

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

### Step 1: Set up the inequality
We now solve for \( x \) by simplifying the inequality. First, subtract 30,000 from both sides:

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

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

### Step 2: Solve the quadratic inequality
We need to solve the quadratic equation:

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

Using the quadratic formula, where \( a = 1 \), \( b = -370 \), and \( c = 33000 \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \( a \), \( b \), and \( c \):

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

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

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

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

Now, solve for the two possible values of \( x \):

\[
x_1 = \frac{370 + 70}{2} = \frac{440}{2} = 220
\]
\[
x_2 = \frac{370 - 70}{2} = \frac{300}{2} = 150
\]

### Step 3: Interpret the solution
The solution to the quadratic inequality \( x^2 - 370x + 33000 \leq 0 \) lies between the two roots. Therefore, the company should manufacture between 150 and 220 bags to stay within the budget.

### Final Answer:
The company should manufacture between **150 and 220 bags** per week without overspending.

---

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

Here are some related questions you might explore:
1. What happens if the cost function changes to \( x^2 - 400x + 64000 \)?
2. How would increasing the budget to 35,000 pesos affect the solution?
3. What if the production cost includes a fixed maintenance fee of 5,000 pesos weekly?
4. How would solving this problem graphically look?
5. Can we use this method for non-quadratic cost functions?

**Tip:** The discriminant (\( b^2 - 4ac \)) helps you determine if a quadratic equation has real solutions!

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

Learn how to solve a quadratic inequality involving manufacturing costs for a company. This problem explores algebraic concepts such as quadratic equations and inequalities to find how many bags a company can produce without exceeding a 30,000 pesos budget. Suitable for Grades 9-10.

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