Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Chapter 10, Problem 13P

Investing in Bonds A woman wishes to invest $12,000 in three types of bonds: municipal bonds paying 7% interest per year, bank certificates paying 8%, and high-risk bonds paying 12%. For tax reasons she wants the amount invested in municipal bonds to be at least three times the amount invested in bunk certificates. To keep her level of risk manage-able, she will invest no more than $2000 in high-risk bonds. How much should she invest in each type of bond to maximize her annual interest yield? [Hint: Let x = amount in municipal bonds and y = amount in bunk certificates. Then the amount in high-risk bonds will be 12,000 − x − y.]

Expert Solution & Answer
Check Mark
To determine

To find: The amount money a women invest in the municipal bond, bank certificate bond and high-risk bond for the maximum interest.

Answer to Problem 13P

Women invest $7500 in municipal bonds, $2500 in the certificate bond and $2000 in high risk bonds.

Explanation of Solution

Given:

The total amount of money is $12000 and invest in the three types of bonds the municipal bond, bank certificate bond and high-risk bond.

The municipal bond pays 7% interest per year, bank certificate pays 8% interest per year and high risk bond pays 12% interest per year.

The amount of money a women invested in municipal bond is three times the invested in bank certificates.

Women invest no more than $2000 in high risk bonds.

Calculation:

Suppose the amount invested in the municipal bond is x and the amount invested in the bank certificate bond is y for the maximum interest.

Then the amount invested in the high risk bond is (12000−x−y) .

Use the given information to make the inequalities and the objective function for the feasible region.

The required information is shown in the table below.

municipal bond Bank certificate bond high-risk bond
Amount of money x y (12000−x−y)

The objective function is,

I=7x100+8y100+12(12000−x−y)100=12(12000−x−y)+7x+8y100=1440−5x100−4y100 ,

The constraint to get the feasible region has shown below.

x+y≥10000x+y≤12000x≥3yx≥0,y≥0}

Now, take the equalities of the above inequalities,

x+y=10000 (1)

And,

x+y=12000 (2)

And,

x=3y (3)

And,

x≥0,y≥0

Substitute 3y for x in the equation (1)

3y+y=100004y=10000y=2500

Substitute 2500 for y in equation (3),

x=3y=3⋅2500=7500

The intersection point is (x,y)=(7500,2500) .

Substitute 3y for x in the equation (2).

3y+y=120004y=12000y=3000

Substitute 3000 for y in equation (3),

x=3y=3⋅3000=9000

Now, draw the graph of the above equations,

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 10, Problem 13P

Figure (1)

The vertices which lies in the feasible region is shown below.

(7500,2500),(10000,0),(12000,0),(9000,3000)

Substitute the 7500 for x and 2500 for y in the objective function I=1440−5x100−4y100 ,

I=1440−5⋅7500100−4⋅2500100=1440−37500100−10000100=1440−375−100=965

Substitute the 10000 for x and 0 for y in the objective function I=1440−5x100−4y100 ,

I=1440−5⋅10000100−4⋅0100=1440−50000100=1440−500=940

Substitute the 12000 for x and 0 for y in the objective function I=1440−5x100−4y100 ,

I=1440−5⋅12000100−4⋅0100=1440−60000100=1440−600=960

Substitute the 9000 for x and 3000 for y in the objective function I=1440−5x100−4y100 ,

I=1440−5⋅9000100−4⋅3000100=1440−45000100−12000100=1440−450−120=870

So, all the satisfies these vertices are shown in the table below.

Vertices I=1440−5x100−4y100
(7500,2500) 965(Maximum)
(10000,0) 940
(9000,3000) 840
(12000,0) 960

The maximum interest rate is 965 at the investment $7500 in municipal bonds, $2500 in the certificate bond. Then the amount of money invested in high risk bond is,

12000−x−y=12000−7500−2500=12000−10000=2000

Thus, a women invest $7500 in municipal bonds, $2500 in the certificate bond and $2000 in high risk bonds at maximum interest for the objective function I=1440−5x100−4y100 .

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Chapter 10, Problem 14P

Chapter 10 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 10.1 - The system of equations {2x+3y=75xy=9 is a system...Ch. 10.1 - Prob. 2ECh. 10.1 - A system of two linear equations in two variables...Ch. 10.1 - The following is a system of two linear equations...Ch. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Prob. 10E
Ch. 10.1 - Prob. 11ECh. 10.1 - Prob. 12ECh. 10.1 - Prob. 13ECh. 10.1 - Prob. 14ECh. 10.1 - Prob. 15ECh. 10.1 - Prob. 16ECh. 10.1 - Prob. 17ECh. 10.1 - Prob. 18ECh. 10.1 - Prob. 19ECh. 10.1 - Prob. 20ECh. 10.1 - Prob. 21ECh. 10.1 - Prob. 22ECh. 10.1 - Prob. 23ECh. 10.1 - Prob. 24ECh. 10.1 - Prob. 25ECh. 10.1 - Prob. 26ECh. 10.1 - Prob. 27ECh. 10.1 - Prob. 28ECh. 10.1 - Prob. 29ECh. 10.1 - Prob. 30ECh. 10.1 - Prob. 31ECh. 10.1 - Prob. 32ECh. 10.1 - Prob. 33ECh. 10.1 - Prob. 34ECh. 10.1 - Prob. 35ECh. 10.1 - Prob. 36ECh. 10.1 - Prob. 37ECh. 10.1 - Prob. 38ECh. 10.1 - Prob. 39ECh. 10.1 - Prob. 40ECh. 10.1 - Prob. 41ECh. 10.1 - Prob. 42ECh. 10.1 - Prob. 43ECh. 10.1 - Prob. 44ECh. 10.1 - Prob. 45ECh. 10.1 - Prob. 46ECh. 10.1 - Prob. 47ECh. 10.1 - Prob. 48ECh. 10.1 - Prob. 49ECh. 10.1 - Prob. 50ECh. 10.1 - Prob. 51ECh. 10.1 - Prob. 52ECh. 10.1 - Prob. 53ECh. 10.1 - Prob. 54ECh. 10.1 - Prob. 55ECh. 10.1 - Prob. 56ECh. 10.1 - Prob. 57ECh. 10.1 - Prob. 58ECh. 10.1 - Value of Coins A man has 14 coins in his pocket,...Ch. 10.1 - Admission Fees The admission fee at an amusement...Ch. 10.1 - Gas Station A gas station sells regular gas for...Ch. 10.1 - Fruit Stand A fruit stand sells two varieties of...Ch. 10.1 - Airplane Speed A man flies a small airplane from...Ch. 10.1 - Boat Speed A boat on a river travels downstream...Ch. 10.1 - Nutrition A researcher performs an experiment to...Ch. 10.1 - Coffee Blends A customer in a coffee shop...Ch. 10.1 - Mixture Problem A chemist has two large containers...Ch. 10.1 - Mixture Problem A biologist has two brine...Ch. 10.1 - Investments A woman invests a total of 20,000 in...Ch. 10.1 - Investments A man invests his savings in two...Ch. 10.1 - Prob. 71ECh. 10.1 - Prob. 72ECh. 10.1 - Prob. 73ECh. 10.1 - Area of a Triangle Find the area of the triangle...Ch. 10.1 - Prob. 75ECh. 10.2 - These exercises refer to the following system:...Ch. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - Prob. 5ECh. 10.2 - Prob. 6ECh. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Prob. 9ECh. 10.2 - 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Prob. 44RECh. 10 - Prob. 45RECh. 10 - Prob. 46RECh. 10 - Prob. 47RECh. 10 - Prob. 48RECh. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Prob. 55RECh. 10 - Prob. 56RECh. 10 - Prob. 57RECh. 10 - Prob. 58RECh. 10 - Prob. 59RECh. 10 - Prob. 60RECh. 10 - Prob. 61RECh. 10 - Prob. 62RECh. 10 - Prob. 63RECh. 10 - Prob. 64RECh. 10 - Prob. 65RECh. 10 - Prob. 66RECh. 10 - Prob. 67RECh. 10 - Prob. 68RECh. 10 - Prob. 69RECh. 10 - Prob. 70RECh. 10 - Prob. 71RECh. 10 - Prob. 72RECh. 10 - Prob. 73RECh. 10 - Prob. 74RECh. 10 - Prob. 75RECh. 10 - Prob. 76RECh. 10 - Prob. 77RECh. 10 - Prob. 78RECh. 10 - Prob. 79RECh. 10 - Prob. 80RECh. 10 - Prob. 81RECh. 10 - Prob. 82RECh. 10 - Prob. 83RECh. 10 - Prob. 84RECh. 10 - Prob. 85RECh. 10 - Prob. 86RECh. 10 - Prob. 87RECh. 10 - Prob. 88RECh. 10 - Prob. 89RECh. 10 - Prob. 90RECh. 10 - Prob. 91RECh. 10 - Prob. 92RECh. 10 - Prob. 93RECh. 10 - Prob. 94RECh. 10 - Prob. 95RECh. 10 - Prob. 96RECh. 10 - Prob. 97RECh. 10 - Prob. 98RECh. 10 - Prob. 99RECh. 10 - Prob. 100RECh. 10 - Prob. 101RECh. 10 - Prob. 102RECh. 10 - Prob. 103RECh. 10 - Prob. 104RECh. 10 - Prob. 105RECh. 10 - Prob. 106RECh. 10 - Prob. 107RECh. 10 - Prob. 108RECh. 10 - Prob. 109RECh. 10 - Prob. 110RECh. 10 - Prob. 111RECh. 10 - Prob. 112RECh. 10 - Prob. 1TCh. 10 - Prob. 2TCh. 10 - Prob. 3TCh. 10 - Prob. 4TCh. 10 - Prob. 5TCh. 10 - Prob. 6TCh. 10 - Prob. 7TCh. 10 - Anne, Barry, and Cathy enter a coffee shop. Anne...Ch. 10 - Prob. 9TCh. 10 - Prob. 10TCh. 10 - Only one of the following matrices has an inverse....Ch. 10 - Prob. 12TCh. 10 - Prob. 13TCh. 10 - Prob. 14TCh. 10 - Find the maximum and minimum values of the given...Ch. 10 - Prob. 2PCh. 10 - Prob. 3PCh. 10 - Find the maximum and minimum value of the given...Ch. 10 - Making Furniture A furniture manufacturer makes...Ch. 10 - A Housing Development A housing contractor has...Ch. 10 - Hauling Fruit A trucker hauls citrus fruit from...Ch. 10 - Manufacturing Calculators A manufacturer of...Ch. 10 - Shipping Televisions An electronics discount chain...Ch. 10 - Prob. 10PCh. 10 - Packaging Nuts A confectioner sells two types of...Ch. 10 - Feeding Lab Rabbits A biologist wishes to feed...Ch. 10 - Investing in Bonds A woman wishes to invest 12,000...Ch. 10 - Prob. 14PCh. 10 - Business Strategy A small software company...Ch. 10 - Feasible Region All parts of this problem refer to...
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