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 770 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 2,100 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 $9700 per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at $7900 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.)

high-income patients 1Your answer is incorrect.
homeless patients 2Your answer is incorrect.
profit

h=number of homeless patients

H=number of high-income patients

Constraints:
1. h≥770 (minimum)
2. H≤900 (maximum)
3. H≤2h (maximum ratio)
Capacity
4. H+h=2100
Z=profit = 9700h + 7900H

Obviously the objective function Z is maximized when h is at its maximum possible.

The response can be found by inspection, namely h=1400, H=700, which satisfies all 4 constraints.

If the answer is not obvious, try equal numbers (1050 each) and change for as many homeless as possible without violating any of the constraints.

I had these as the equations but how do u set them up to find how many of each to maximise profit

It depends on what your teacher expects you to know. Have you done simplex method, graphics, or trial and error?

You can plot the constraints on a graph as a line, and shade the non-feasible regions (usually on one side of the line). After all four constraints have been constructed and the appropriate sides shaded, the remaining unshaded region is the feasible region.

Evaluate the objective function at each of the corners of the feasible polygon, and find the maximum.

P.S.
I left out two other (trivial) constraints, namely h≥0 and H≥0.

noo

Sorry, there was a mistake in the formulation of the constraints.

The line H≤2h should have read h≤2H to reflect "the number of homeless patients should not exceed twice the number of high-income patients"
So a summary of the constraints now reads:

1. h≥770 (minimum)
2. H≥0 (minimum H)
3. H≤900 (maximum)
4. H≥h/2 (minimum ratio for H, maximum ratio for h)
5. H+h=2100 (capacity)

Z=profit = 9700h + 7900H
Z is to be evaluated at corners of the polygon.

Check:
(Broken Link Removed)
Legend:
red - max value of H
blue - capacity of 2100
green - min. ratio H/h

We see that the corners of the polygon of feasibility are at
(770,900), (1200,900), (1400,700), (770,385).

Evaluate the objective function Z at these points and make your choice.

To solve this problem, we can use a linear programming approach.

First, let's define our variables:
- Let H represent the number of homeless patients.
- Let I represent the number of high-income patients.

Now, let's create our objective function. The hospital's profit is determined by the number of homeless and high-income patients. The profit per homeless patient is $9700 per month, while the profit per high-income patient is $7900 per month. Thus, our objective function for profit (P) is:
P = 9700H + 7900I

Next, let's define our constraints based on the given information and requirements:
1. The overall capacity of the hospital is 2100 patients. Therefore, the total number of patients must be less than or equal to 2100:
H + I ≤ 2100

2. State regulations require at least 770 homeless patients and no more than 900 high-income patients to qualify for subsidies:
770 ≤ H ≤ 900
0 ≤ I

3. The board of directors insists that the number of homeless patients should not exceed twice the number of high-income patients:
H ≤ 2I

Now, we can solve this linear programming problem to maximize profit.

The possible values for the number of homeless patients (H) and high-income patients (I) lie within the feasible region defined by the constraints.

To solve the problem, we can use optimization software or graphing techniques. By solving this problem using optimization software or graphing the constraints, we can find the values of H and I that maximize the profit (P).