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The image you uploaded contains several mathematical problems, primarily related to algebra and geometry. I'll provide a solution and explanation for **Problem 32** as an example:

### Problem 32:
*Perusahaan ABC harus mengeluarkan biaya \(400 + 5x + \frac{x^2}{6} - 4\) rupiah untuk membuat \(x\) buah mainan kompor yang dijual seharga 6 rupiah sebuah.*

The task has three parts:

#### (a) **Find the formula for \(P(x)\), the total profit of making \(x\) stoves.**

Profit is given by the formula:
\[
P(x) = \text{Revenue} - \text{Cost}
\]
The **revenue** is the amount earned from selling \(x\) stoves at a price of 6 rupiah each:
\[
\text{Revenue} = 6x
\]
The **cost** to produce \(x\) stoves is given as:
\[
\text{Cost} = 400 + 5x + \frac{x^2}{6} - 4
\]
Therefore, the profit \(P(x)\) is:
\[
P(x) = 6x - (400 + 5x + \frac{x^2}{6} - 4)
\]
Simplifying this:
\[
P(x) = 6x - 400 - 5x - \frac{x^2}{6} + 4
\]
\[
P(x) = x - \frac{x^2}{6} - 396
\]

#### (b) **Find \(P(200)\) and \(P(1000)\):**

To find \(P(200)\) and \(P(1000)\), substitute \(x = 200\) and \(x = 1000\) into the formula for \(P(x)\).

For \(P(200)\):
\[
P(200) = 200 - \frac{200^2}{6} - 396
\]

For \(P(1000)\):
\[
P(1000) = 1000 - \frac{1000^2}{6} - 396
\]

Let me calculate these two values for you.The values of the profit function are:

- \( P(200) = -6862.67 \)
- \( P(1000) = -166062.67 \)

These negative values indicate a loss at both production levels of 200 and 1000 stoves.

#### (c) **How many stoves must ABC produce to break even (no profit or loss)?**

To find the break-even point, set \(P(x) = 0\) and solve for \(x\):

\[
x - \frac{x^2}{6} - 396 = 0
\]

This is a quadratic equation. Let me solve it for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can solve the quadratic equation by hand using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where the equation is in the form \( ax^2 + bx + c = 0 \). Here, \( a = -\frac{1}{6}, b = 1, \) and \( c = -396 \).

Would you like assistance in solving this manually?

### Follow-up Questions:
1. How does the quadratic formula help solve this problem?
2. Why is the profit negative for the given values of \( x \)?
3. Can we use any other method to find the break-even point?
4. What would happen if the cost function changes over time?
5. How does the break-even point affect production planning?

**Tip:** Always recheck the units when solving cost and revenue problems to ensure they align.

Perusahaan ABC harus mengeluarkan biaya 400 + 5x + x^2/6 - 4 rupiah untuk membuat x buah mainan kompor yang dijual seharga 6 rupiah sebuah. Cari rumus keuntungan total dan hitung jumlah kompor untuk impas.

Learn how to calculate the profit and break-even point for a production cost function that includes quadratic terms. This problem focuses on algebraic concepts and the use of the quadratic formula to solve for x, with a detailed explanation suitable for high school students.

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