Need a system of inequalities & objective function

Question

Trenton, Michigan, a small community, is trying to establish a public transportation system of large and small vans. It can spend no more than $100,000 for both sizes of vehicles and no more than $500 per month for maintenance. The community can purchase a small van for $10,000 and maintain it for $100 per month. The large vans cost $20,000 each and can be maintained for $75 per month. Each large van carries a maximum of 15 passengers, and each small van carries a maximum of 7 passengers. How many of each van should they purchase? Find the maximum number of passengers.

Answer 1

We will start by assigning variables for the number of small vans and number of large vans which are what we are trying to determine.

Let x = Number of small vans and y = Number of large vans.

Now let's look at the constraints.

1. Purchase cost <= 100000 which means

10000 x + 20000 y <= 100000 which may be simplified as

x + 2y <= 10

1. Maintenance cost <= 500 meaning

100x + 75 y <= 500 which may be simplified as

4x + 3y <= 20

We can add two other inequalities as x>= 0 and y >= 0 as the number of vans cannot be negative.

The objective function would be to maximize the number of passengers.

which would be 7x + 15 y

If we draw the 4 inequalities, The vertices of the bounded region would be (0,0), (5,0) , (0,5) and the intersection of the lines x + 2y = 10 and 4x + 3y = 20 which is (2,4)

The value of Objective function at these vertices will be 0, 35, 75 and 74 respectively for (0,0), (5,0), (0,5) and (2,4)

The point (0,5) is giving us the maximum passengers of 75. Hence, they should purchase 5 large vans and no small vans.

Note: I tried to record a video answer, but I was facing technical difficulties.