https://file.aiquickdraw.com/mathbot/file/1729274970916wo2sz9sk.jpg

Let's analyze the problem.

The profit function given is:

\[
P(x) = -10x^2 + 1620x - 49000
\]

To maximize the monthly rental profit, we need to find the value of $ x $ that gives the maximum point of this quadratic function. Since the coefficient of $ x^2 $ is negative ($-10$), the parabola opens downward, and the maximum profit occurs at the vertex of the parabola.

### Step 1: Finding the Vertex
The vertex of a quadratic function $ ax^2 + bx + c $ occurs at:

\[
x = -\frac{b}{2a}
\]

Here, $ a = -10 $ and $ b = 1620 $. Substituting these values, we get:

\[
x = -\frac{1620}{2(-10)} = \frac{1620}{20} = 81
\]

Therefore, the maximum profit occurs when 81 units are rented out. This is consistent with the answer already provided.

### Step 2: Finding the Maximum Profit
We now need to find the maximum profit by substituting $ x = 81 $ into the profit function $ P(x) $:

\[
P(81) = -10(81)^2 + 1620(81) - 49000
\]

Calculating each term:

1. $-10 \cdot 81^2 = -10 \cdot 6561 = -65610$
2. $1620 \cdot 81 = 131220$

Thus:

\[
P(81) = -65610 + 131220 - 49000 = 16610
\]

The maximum monthly profit is **$16,610**, not $32,220 as mentioned in the image.

### Summary:
- **Units to rent out for maximum profit:** 81
- **Maximum monthly profit:** $16,610

Would you like further details or explanations on any step? 

Here are 5 related questions to explore this topic further:

1. How can we find the maximum point of a quadratic function if it does not open downward?
2. What happens to the profit function if the cost per unit changes?
3. How does the coefficient of $ x^2 $ affect the shape of the parabola?
4. Can the profit function have more than one maximum point?
5. How do real-world constraints (like a limit on the number of units) affect the calculation of maximum profit?

**Tip:** For quadratic functions, always look for the vertex to find maximum or minimum values.

To maximize the monthly rental profit, how many units should be rented out? What is the maximum monthly profit (in dollars) realizable?

Learn how to maximize the monthly rental profit of an apartment complex with 100 units using the quadratic function P(x) = -10x^2 + 1620x - 49000. Find the number of units to rent for maximum profit and the corresponding profit amount. Suitable for high school students.

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