The Megabuck Hospital Corporation is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 730 homeless patients and no more than 900 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,400 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $9,500 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7,700 per month. How many of each type of patient should it house in order to maximize profit? (If an answer does not exist, enter DNE.)
high-income patients
homeless patients
profit			$

Let's solve this problem step by step.

### 1. Define the Variables
Let:
- $ x $ be the number of high-income patients.
- $ y $ be the number of homeless patients.

### 2. Write Down the Constraints
We are given the following constraints:
1. The total number of patients is 2,400:
   \[
   x + y = 2400
   \]
2. The minimum number of homeless patients is 730:
   \[
   y \geq 730
   \]
3. The maximum number of high-income patients is 900:
   \[
   x \leq 900
   \]
4. The number of homeless patients should not exceed twice the number of high-income patients:
   \[
   y \leq 2x
   \]

### 3. Objective Function
The objective is to maximize the profit, which is given by:
\[
\text{Profit} = 9500y + 7700x
\]

### 4. Solve the System
We need to maximize the profit under the given constraints.

#### Substituting $ y = 2400 - x $ into the constraints:
1. $ y \geq 730 $:
   \[
   2400 - x \geq 730 \implies x \leq 1670
   \]
2. $ y \leq 2x $:
   \[
   2400 - x \leq 2x \implies 2400 \leq 3x \implies x \geq 800
   \]
3. The constraint $ x \leq 900 $ still holds.

Thus, $ x $ must satisfy:
\[
800 \leq x \leq 900
\]

#### Substitute $ y = 2400 - x $ in the profit function:
\[
\text{Profit} = 9500(2400 - x) + 7700x
\]
Simplifying the profit function:
\[
\text{Profit} = 9500 \times 2400 - 9500x + 7700x
\]
\[
\text{Profit} = 22800000 - 1800x
\]

Since the profit function $ 22800000 - 1800x $ is a linear decreasing function of $ x $, to maximize the profit, we should minimize $ x $ within the range $ 800 \leq x \leq 900 $.

#### Conclusion
To maximize the profit:
- Set $ x = 800 $.
- $ y = 2400 - 800 = 1600 $.

So, the optimal number of patients is:
- **High-income patients**: $ 800 $
- **Homeless patients**: $ 1600 $

### 5. Calculate the Maximum Profit
Substitute $ x = 800 $ and $ y = 1600 $ into the profit function:
\[
\text{Profit} = 9500 \times 1600 + 7700 \times 800
\]
\[
\text{Profit} = 15200000 + 6160000 = 21360000
\]

Thus, the maximum profit is $ \text{\$21,360,000} $.

### Final Answer
- High-income patients: **800**
- Homeless patients: **1600**
- Maximum Profit: **\$21,360,000**

Would you like more details on the calculations, or do you have any questions?

Here are some related questions:
1. What would be the profit if the hospital only houses homeless patients?
2. How would the profit change if the capacity of the hospital increased by 100 patients?
3. What if the profit per homeless patient increased to $10,000, how would the optimal solution change?
4. What is the minimum number of high-income patients needed to still qualify for the state subsidy?
5. How would the problem change if the number of high-income patients had no upper limit?

**Tip:** Always check if your solution satisfies all given constraints when solving optimization problems.

The Megabuck Hospital Corporation is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 730 homeless patients and no more than 900 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,400 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of $9,500 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7,700 per month. How many of each type of patient should it house in order to maximize profit? (If an answer does not exist, enter DNE.)

Explore how to use linear programming to maximize the profit for a state-subsidized nursing home that houses homeless and high-income patients. This problem involves constraints related to patient numbers and state regulations, making it ideal for students in Grades 11-12 studying optimization problems.

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