A company spends as much as x2 - 370x + 63000 pesos to manufacture x number of bags in a wehek. 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? Show the solution

We are given the cost function:

\[
C(x) = x^2 - 370x + 63,000
\]

where \(C(x)\) represents the cost of manufacturing \(x\) bags, and we know the company wants the cost to be at most 30,000 pesos. This means we must solve the inequality:

\[
x^2 - 370x + 63,000 \leq 30,000
\]

### Step 1: Set up the inequality
Subtract 30,000 from both sides to form a standard quadratic inequality:

\[
x^2 - 370x + 63,000 - 30,000 \leq 0
\]
\[
x^2 - 370x + 33,000 \leq 0
\]

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

\[
x^2 - 370x + 33,000 = 0
\]

We'll use the quadratic formula:

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

where \(a = 1\), \(b = -370\), and \(c = 33,000\).

First, calculate the discriminant:

\[
\Delta = b^2 - 4ac = (-370)^2 - 4(1)(33,000) = 136,900 - 132,000 = 4,900
\]

Now, calculate \(x\):

\[
x = \frac{-(-370) \pm \sqrt{4,900}}{2(1)} = \frac{370 \pm 70}{2}
\]

This gives us two solutions:

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

### Step 3: Interpret the solution
The quadratic inequality \(x^2 - 370x + 33,000 \leq 0\) means that the values of \(x\) must lie between the two solutions of the quadratic equation. So, the company should produce between 150 and 220 bags per week in order to keep the production cost within the budget.

### Final Answer:
The company should manufacture between **150 and 220 bags** per week to avoid overspending.

---

Would you like further clarification on any of the steps? Here are 5 related questions to explore:

1. How would the answer change if the budget was 35,000 pesos instead of 30,000?
2. Can we graph the cost function and visually confirm the result?
3. What happens if the production cost formula is changed to include a variable cost?
4. How is the quadratic formula derived?
5. How do real-life constraints like fixed costs affect this type of model?

**Tip:** When solving inequalities involving quadratics, always check the sign of the expression within the range between the solutions.

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? Show the solution.

Learn how to solve a quadratic inequality for production costs in a manufacturing scenario. We solve x^2 - 370x + 63,000 ≤ 30,000 to determine how many bags a company can manufacture without exceeding a budget of 30,000 pesos. Suitable for Grades 9-12 students, this solution involves using the quadratic formula and interpreting the results.

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