The Megabuck Hospital Corp. 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 760 homeless patients and no more than 1100 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 1,800 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 $10000 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7600 per month. How many of each type of patient should it house in order to maximize profit? HINT [See Example 3.] (If an answer does not exist, enter DNE.)

To solve this problem, we need to find the number of homeless patients and high-income patients that will maximize the profit for the hospital.

Let's assign variables to represent the number of homeless patients (H) and high-income patients (I) that the hospital will house.

We have the following constraints based on the state regulations and the board of directors' requirement:

Constraint 1: H ≥ 760
Constraint 2: I ≤ 1100
Constraint 3: H ≤ 2I
Constraint 4: H + I ≤ 1800

Now let's consider the profit generated by housing each type of patient:

Profit from each homeless patient = $10,000
Profit from each high-income patient = $7,600

We want to maximize the total profit, so our objective function is:

Total Profit = (Profit per homeless patient) * H + (Profit per high-income patient) * I

Total Profit = 10,000H + 7,600I

To solve this problem using linear programming, we will follow these steps:

Step 1: Graph the feasible region based on the constraints.
Step 2: Identify the corner points of the feasible region.
Step 3: Evaluate the objective function at each corner point.
Step 4: Select the corner point that maximizes the objective function.

Let's solve it step by step:

Step 1: Graph the feasible region based on the constraints.

Since we have four constraints, we will graph each one individually.

Constraint 1: H ≥ 760
Draw a horizontal line at H = 760.

Constraint 2: I ≤ 1100
Draw a vertical line at I = 1100.

Constraint 3: H ≤ 2I
Draw a line with a slope of 1/2 passing through the origin (0, 0).

Constraint 4: H + I ≤ 1800
Draw a line with a slope of -1 passing through the point (0, 1800).

The feasible region is the area where all these lines intersect or overlap.

Step 2: Identify the corner points of the feasible region.

The corner points are the vertices of the feasible region.

Step 3: Evaluate the objective function at each corner point.

Evaluate the objective function Total Profit = 10,000H + 7,600I at each corner point.

Step 4: Select the corner point that maximizes the objective function.

Choose the corner point with the maximum total profit.

By following these steps, you can find the optimal solution for the given problem and determine the number of homeless patients and high-income patients that should be housed to maximize profit.