Question

Joey is buying plants for his garden. He wants to have at least twice as many flowering plants as nonflowering plants and a minimum of 36 plants in his garden. Flowering plants sell for $8, and nonflowering plants sell for $5. Joey wants to purchase a combination of plants that minimizes cost. Let x represent the number of flowering plants and y represent the number of nonflowering plants.

What are the vertices of the feasible region for this problem?
(0, 0), (0, 36), (24, 12)
(0, 36), (24, 12)
(0, 36), (24, 12), (36, 0)
(24, 12), (36, 0)

Answer 1

Answer: (24, 12), (36, 0)

Explanation:

Let x be the number of flowering plants and y be the number of non- flowering plants.

According to the question, we need to minimize the cost of plants.

`Minimize:8x+5y`

Subject to the constraints,

`2y\leq\ x\\x+y\geq36`

To find the feasible region find the points of the equation to plot it on graph.

For the first equation
`2y=x` , at x=0 y=0 and at x=4, y=2

For the second equation
`x+y=36` , at x=0 y=36 and at x=36, y=0

Thus points for eq (1) are (0,0) and (4,2) and points for equation (2) are (0,36) and (36,0).

Now, plot it on graph, we get the shaded feasible region as shown in the graph.

and we can see the vertices of the feasible region = (24, 12), (36, 0)

Answer 2

(24, 12) and (36, 0). The least amount of flowering plants occurs when x=2y, and the largest amount occurs when y=0. These two points satisfy both conditions and both sum to 36.