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 14P
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.

Expert Solution & Answer
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Answer to Problem 14P

Women invest $6750 in municipal bonds, $2250 in the certificate bond and $3000 in high risk bonds to get the maximum interest.

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 $3000 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=90004y=9000y=2250

Substitute 2250 for y in equation (3),

x=3y=3⋅2250=6750

Thus, the intersection point is (x,y)=(6750,2250) .

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 14P

Figure (1)

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

(6750,2250),(9000,3000),(12000,0),(9000,0)

Substitute the 6750 for x and 2250 for y in the objective function I=1440−5x100−4y100 ,

I=1440−5⋅6750100−4⋅2250100=1440−33750100−9000100=1440−337.5−90=1012.5

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

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

I=1440−5⋅9000100−4⋅0100=1440−45000100−0=1440−450=990

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

Vertices I=1440−5x100−4y100
(6750,2250) 1012.5(Maximum)
(9000,0) 870
(9000,0) 990
(12000,0) 960

The maximum interest rate is 1012.5 at the investment $6750 in municipal bonds, $2250 in the certificate bond. Then the amount of money invested in high risk bond is,

12000−x−y=12000−6750−2250=12000−9000=3000

Thus, a women invest $6750 in municipal bonds, $2250 in the certificate bond and $3000 in high risk bonds at maximum interest for the objective function I=1440−5x100−4y100 .

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Chapter 10 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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