Graph the system of constraints and find the value of x and y that maximize the objective function

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A manufacturer produces two types of bottled coffee drinks: cappuccinos and cafés au lait. Each bottle of cappuccino requires 6 ounces of coffee and 2 ounces of milk and earns a profit of $0.40. Each bottle of café au lait requires 4 ounces of coffee and 4 ounces of milk and earns a profit of $0.50. The manufacturer has 720 ounces of coffee and 400 ounces of milk available for production each day. To meet demand, the manufacturer must produce at least 80 coffee drinks each day. Let x = the number of cappuccino bottles and y = the number of café au lait bottles.

Identify the constraints on the system other than
x ≥ 0 and y ≥ 0.

1st
3rd
4th

Complete the objective function.

P =
x +
y

0.40
0.50

A system has the following constraints:
x + y ≥ 80
3x + 2y ≤ 360
x + 2y ≤ 200 x ≥ 0
y ≥ 0

Which graph represents the feasible region for the system?

C

The vertices of the feasible region represented by a system are (0, 100), (0, 80), (80, 60), (80, 0), and (120, 0).

What are the minimum and maximum values of the objective function F = 8x + 5y?
Minimum:

Maximum:

400
960

A printing company orders paper from two different suppliers. Supplier X charges $25 per case. Supplier Y charges $20 per case. The company needs to order at least 45 cases per day to meet demand and can order no more than 30 cases from Supplier X. The company needs no more than 2 times as many cases from Supplier Y as from Supplier X. Let x = the number of cases from Supplier X and y = the number of cases from Supplier Y.

Complete the constraints on the system.

y less than equal to x
x + y greater than equal to
x less than equal to

2
45
30

The printing company wants to minimize costs. What is the objective function?

D

The graph represents the feasible region for the system:

y es001-1.jpg 2x x + y es001-2.jpg 45
x es001-3.jpg 30

Minimize the objective function P = 20x + 16y.

The minimum value =
and occurs when x =
and y =
.

780
15
30

Given constraints:
x es002-1.jpg 0, y es002-2.jpg 0, 2x + 2y es002-3.jpg 4, x + y es002-4.jpg 8

Explain the steps for maximizing the objective function P = 3x + 4y.

Graph the inequalities given by the set of constraints. Find points where the boundary lines intersect to form a polygon. Substitute the coordinates of each point into the objective function and find the one that results in the largest value.

A company produces two products, A and B. At least 30 units of product A and at least 10 units of product B must be produced. The maximum number of units that can be produced per day is 80. Product A yields a profit of $15 and product B yields a profit of $8. Let a = the number of units of product A and b = the number of units of product B.

What objective function can be used to maximize the profit?
P =
a +
b

15
8

The vertices of the feasible region are (70, 10), (30, 10), and (30, 50). To maximize the profit, the company should produce units of product A and units of product B. The maximum profit is $

70
10
1130

The graph shows the feasible region for the system with constraints:
y mr001-1.jpg 15 x + y mr001-2.jpg 25 x + 2y mr001-3.jpg 30
What are the vertices of the feasible region? Check all of the boxes that apply.

(0,15)
(10,15)
(20,5)

What is the minimum value of the objective function C = 4x + 9y?
C =

125

Given the system of contstraints:

y ≥ 2x x + y ≤ 14 y ≥ 1
5x + y ≥ 14 x + y ≥ 9
Which region represents the graph of the feasible region for the given constraints?

A

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Verified questions

ADVANCED MATH

If $\bigcap_{\alpha \in I} A_{\alpha}=\bigcup_{\alpha \in I} A_{\alpha}$, what do you think can be said about the relationships between the sets $A_{\alpha}$?

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ADVANCED MATH

a. After playing the game twice, what is the probability that 1 will have $3? How about$2? b. After playing the game three times, what is the probability that 1 will have $2?

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ADVANCED MATH

Alcoa produces 100-, 200-, and 300-foot-long aluminum ingots for customers. This week’s demand for ingots is shown in Table 115. Alcoa has 4 furnaces in which ingots can be produced. During a week, each furnace can be operated for 50 hours. Because ingots are produced by cutting long strips of aluminum, longer ingots take less time to produce than shorter ingots. If a furnace is devoted completely to producing one type of ingot; the number it can produce in a week is shown in Table 116. For example, furnace 1 could produce 350 300-foot ingots per week. The material in an ingot costs $10 per foot. If a customer wants a 100- or 200-foot ingot, then she will accept an ingot of that length or longer. How can Alcoa minimize the material costs incurred in meeting required weekly demands? TABLE 115: $$ \begin{matrix} \text{Ingot (ft)} & \text{Demand}\\\text{100} & \text{700}\\ \text{200} & \text{300}\\ \text{300} & \text{150}\\ \end{matrix} $$ TABLE 116: $$ \begin{matrix} \text{ } & \text{Ingot Length}\\ \text{Furnace} & \text{100'} & \text{200'} & \text{300'}\\ \text{1} & \text{230} & \text{340} & \text{350}\\ \text{2} & \text{230} & \text{260} & \text{280}\\ \text{3} & \text{240} & \text{300} & \text{310}\\ \text{4} & \text{200} & \text{280} & \text{300}\\\end{matrix} $$

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ADVANCED MATH

A company produces A, B, and C and can sell these products in unlimited quantities at the following unit prices: A, $10; B,$56; C, $100. Producing a unit of A requires 1 hour of labor; a unit of B, 2 hours of labor plus 2 units of A; and a unit of C, 3 hours of labor plus 1 unit of B. Any A that is used to produce B cannot be sold. Similarly, any B that is used to produce C cannot be sold. A total of 40 hours of labor are available. Formulate an LP to maximize the company’s revenues.

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