Explanation Given: Deb has $100,000 that she wants to invest. She will spend at least $40,000 on government securities and she wants half of her inheritance to go to bonds and mutual funds. Government securities have an initial fee of 2%, municipal bonds have an initial fee of 1%, and mutual funds have an initial fee of 3%. Government securities earn 7% per year, municipal bonds earn 6% per year, and mutual funds earns 10% per year. Explanation: Let there be $ x 1 be invested in government securities, $ x 2 be invested in municipal bonds, and $ x 3 be invested in mutual funds. Then, form a system of linear inequalities using the provided statement. The system of inequalities is x 1 + x 2 + x 3 ≤ 100 , 000 x 1 ≥ 40 , 000 x 2 + x 3 ≥ 50 , 000 0.02 x 1 + 0.01 x 2 + 0.03 x 3 ≤ 2400 The objective function to be maximized is z = 0.07 x 1 + 0.06 x 2 + 0.1 x 3 . Introduce slack variables s 1 and s 4 . Introduce surplus variable s 2 and s 3 . The above system of linear inequalities becomes the following system of linear equations: x 1 + x 2 + x 3 + s 1 = 100 , 000 x 1 − s 2 = 40 , 000 x 2 + x 3 − s 3 = 50 , 000 0.02 x 1 + 0.01 x 2 + 0.03 x 3 + s 4 = 2400 z − 0.07 x 1 − 0.06 x 2 − 0.1 x 3 = 0 Also, x 1 , x 2 , x 3 ≥ 0 This produces the following tableau: [ x 1 x 2 x 3 s 1 s 2 s 3 s 4 z 1 1 1 1 0 0 0 0 100 , 000 1 0 0 0 − 1 0 0 0 40 , 000 0 1 1 0 0 − 1 0 0 50 , 000 0.02 0.01 0.03 0 0 0 1 0 2400 − 0.07 − 0.06 − 0.1 0 0 0 0 1 0 ] _ The solution of this tableau gives s 2 = − 100 , 000 , which is an unwanted solution; hence, we would select a nonnegative entry farthest on the left in the second row. Use ratio test to find the pivot element. 100 , 000 1 = 100 , 000 , 40 , 000 1 = 40 , 000 , 2400 0.02 = 120 , 000 Hence, use 1 as the pivot element as it gives the least ratio. Use the first element of the second row as the pivot element. Use row operation R 1 − R 2 → R 1 , R 4 − 0.02 R 2 → R 4 , R 5 + 0.07 R 2 → 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 z 0 1 1 1 1 0 0 0 60 , 000 1 0 0 0 − 1 0 0 0 40 , 000 0 1 1 0 0 − 1 0 0 50 , 000 0 0.01 0.03 0 0.02 0 1 0 1600 0 − 0.06 − 0.1 0 − 0.07 0 0 1 2800 ] _ Since the basic variable s 3 is negative, choose a positive entry farthest to the left in the third row. Use ratio test to find the pivot element in this column. 60 , 000 1 = 60 , 000 , 50 , 000 1 = 50 , 000 , 1600 0

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ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 4.4, Problem 26E

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The amount of money to be invested in government securities, municipal bonds, and mutual funds to maximize the interest income.

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Chapter 4.4, Problem 25E

Chapter 4.4, Problem 27E

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Chapter 4 Solutions

Finite Mathematics (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...

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The amount of money to be invested in government securities, municipal bonds, and mutual funds to maximize the interest income.

Solution Summary: The author explains how Deb makes a maximum return of 8300 when she invests 50,000 on government securities, 5000 on municipal bonds, and 45,000 on mutual funds.

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The amount of money to be invested in government securities, municipal bonds, and mutual funds to maximize the interest income.

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Chapter 4 Solutions