Explanation Given: The total amount of the additive must be at least 10oz per gal of gasoline. For safety reasons, the amount of additive should not exceed 15oz per gal of gasoline. At least 1 4 oz of ingredient I must be used for every ounce of ingredient II. At least 1oz of ingredient III must be used for every ounce of ingredient I. Cost of I, II, and III is $0.30, $0.09, and $0.27per oz, respectively. Explanation: Let the amount of liquid in each batch be x 1 oz of liquid I, x 2 oz of liquid II, and x 3 oz of liquid III. Use the provided statement to form the following system of linear inequalities. x 1 + x 2 + x 3 ≥ 10 x 1 + x 2 + x 3 ≤ 15 1 4 x 1 ≥ x 2 x 3 ≥ x 1 Also, x 1 , x 2 , x 3 ≥ 0 where the objective function to be minimized is z = 0.30 x 1 + 0.09 x 2 + 0.27 x 3 ; hence, assume a function w such that w = − z then the maximum value of w will be equal to the minimum value of the objective function z . Simplify the above linear inequalities and introduce surplus variables s 1 , s 3 , and s 4 , introduce slack variable s 2 to covert the system of linear inequalities into a system of linear equations. x 1 + x 2 + x 3 − s 1 = 10 x 1 + x 2 + x 3 + s 2 = 15 0.25 x 1 − x 2 − s 3 = 0 x 3 − x 1 − s 4 = 0 w + 0.30 x 1 + 0.09 x 2 + 0.27 x 3 = 0 This system of linear equations gives the following simplex tableau: [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 w 1 1 1 − 1 0 0 0 0 10 1 1 1 0 1 0 0 0 15 0.25 − 1 0 0 0 − 1 0 0 0 − 1 0 1 0 0 0 − 1 0 0 0.3 0.09 0.27 0 0 0 0 1 0 ] _ The solution of this tableau gives a negative value of the basic variable s 1 which is not feasible; hence, choose a positive nonzero entry farthest to left in the first row. Use ratio test to find the pivot element in this column. 10 1 = 10 , 15 1 = 15 , 0 0.25 = 0 Hence, the pivot element is 0.25 as the least ratio is obtained for this element. Use row operation 1 0.25 R 2 → R 2 to change the pivot element to 1 and use R 1 − R 3 → R 1 , R 2 − R 3 → R 2 , R 4 + R 3 → R 4 , R 5 − 0.3 R 3 → R 5 To change the remaining elements in the pivot column to 0. [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 w 1 1 1 − 1 0 0 0 0 10 1 1 1 0 1 0 0 0 15 1 − 4 0 0 0 − 4 0 0 0 − 1 0 1 0 0 0 − 1 0 0 0.3 0.09 0.27 0 0 0 0 1 0 ] [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 w 0 5 1 − 1 0 4 0 0 10 0 5 1 0 1 4 0 0 15 1 − 4 0 0 0 − 4 0 0 0 0 − 4 1 0 0 − 4 − 1 0 0 0 1.29 0.27 0 0 1.2 0 1 0 ] _ The solution of this tableau gives a negative value for the basic variable s 1 , choose a positive nonzero entry farthest to left in the first row. Use ratio test to find the pivot element in this column

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 4.4, Problem 32E

To determine

The mixture of the three ingredients that produce the minimum cost of the additive.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.4, Problem 31E

Chapter 4.4, Problem 33E

Students have asked these similar questions

Suppose that a room containing 1300 cubic feet of air is originally free of carbon monoxide (CO). Beginning at time t = 0, cigarette smoke containing 4% CO is introduced into the room at a rate of 0.8 cubic feet per minute. The well-circulated smoke and air mixture is allowed to leave the room at the same rate. Let A(t) represent the amount of CO in the room (in cubic feet) after t minutes. (A) Write the DE model for the time rate of change of CO in the room. Also state the initial condition. dA dt A(0) (B) Solve the IVP to find the amount of CO in the room at any time t > 0. A(t) (C) Extended exposure to a CO concentration as low as 0.00012 is harmful to the human body. Find the time at which this concentration is reached. t= minutes

Newton's Law of Cooling tells us that the rate of change of the temperature of an object is proportional to the temperature difference between the object and its surroundings. This can be modeled by the differential equation dT dt k(TA), where T is the temperature of the object after t units of time have passed, A is the ambient temperature of the object's surroundings, and k is a constant of proportionality. Suppose that a cup of coffee begins at 178 degrees and, after sitting in room temperature of 61 degrees for 12 minutes, the coffee reaches 171 degrees. How long will it take before the coffee reaches 155 degrees? Include at least 2 decimal places in your answer. minutes

Decide whether each limit exists. If a limit exists, estimate its value. 11. (a) lim f(x) x-3 f(x) ↑ 4 3- 2+ (b) lim f(x) x―0 -2 0 X 1234

Chapter 4 Solutions

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 2E Ch. 4.1 - Convert each inequality into an equation by adding...Ch. 4.1 - Prob. 4E Ch. 4.1 - For Exercises 5-8. (a) determine the number of...Ch. 4.1 - Prob. 6E Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - For Exercises 5-8, (a) determine the number of...Ch. 4.1 - Introduce slack variables as necessary, then...Ch. 4.1 - Introduce slack variables as necessary, then write...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The mixture of the three ingredients that produce the minimum cost of the additive.

Solution Summary: The author explains that the minimum cost of the additive is 2.63 when the mixture of three ingredients is used.

Videos

The mixture of the three ingredients that produce the minimum cost of the additive.

Want to see the full answer?

Chapter 4 Solutions