Problem for number 5 linear program

Question

Problem 5

Sherri Enjoy from Shoppe needs to order trees for the Christmas season. She contacts Festive Suppliers, a wholesale store that supplies all sorts of holiday supplies. Festive suppliers sell both real trees and artificial trees.

Holiday Shoppe told Sherri she can order from 20 to 90 real trees and any number up to (and including) 100 artificial trees. Festive Suppliers ordering criteria say they will supply at least 45 to at most 120 trees in order, but they stipulate that the number of artificial trees "must be at least half" of the number of real trees.

Festive Suppliers sells the real trees at $80 per tree and the artificial trees at $160 per tree. The Holiday Shoppe is on a tight budget this season. so Sherri needs to order trees at the "minimum cost" she can, meeting the ordering criteria.

How many of each kind of tree should Sherri order and what will this minimum cost be?

Including

Identify x and y variables.

create your functions formula with these variables.

Write your system of inequalities that make up your constraints.

Graph your system of inequalities.

Show your polygonal feasibility solutions set.

Identify the corner or vertex point of your polygonal set.

use these points in your functions formula to find the maximum value.

give your answer completely in sentences.

Please explain step by step and graphing as well.

Answer 1

Step 1: Identify Variables

Let x be the number of real trees and y be the number of artificial trees Sherri orders.

Step 2: Formulate the Objective Function

The cost of one real tree is $80, and the cost of one artificial tree is $160. The objective is to minimize the cost, so we use the following formula:

Cost = 80x + 160y

Step 3: Formulate Constraints

The number of real trees can be between 20 and 90:

20 ≤ x ≤ 90

The number of artificial trees can be any number up to (and including) 100:

y ≤ 100

The total number of trees ordered must be at least 45 and at most 120:

45 ≤ x + y ≤ 120

The number of artificial trees must be at least half the number of real trees:

y ≥ 0.5x

Step 4: Graph the System of Inequalities

Plotting these constraints on a graph, we get:

Graph

The feasible region is the shaded region in the graph.

Step 5: Identify the Corner or Vertex Points of the Feasible Region

The corner points of the feasible region are (20, 25), (45, 45), (70, 50), (90, 55), (90, 100).

Step 6: Find the Minimum Cost

Substituting these corner points into the objective function, we get:

(20, 25) ⇒ Cost = 80(20) + 160(25) = $6,000

(45, 45) ⇒ Cost = 80(45) + 160(45) = $12,000

(70, 50) ⇒ Cost = 80(70) + 160(50) = $14,000

(90, 55) ⇒ Cost = 80(90) + 160(55) = $15,200

(90, 100) ⇒ Cost = 80(90) + 160(100) = $24,000

Therefore, the minimum cost Sherri can order the trees for is $6,000 by ordering 20 real trees and 25 artificial trees.