Chapter 11.7, Problem 57AYU

Financial Planning A retired couple has up to $\$50,000$ to invest. As their financial adviser, you recommend that they place at least $\$35,000$ in Treasury bills yielding $3\%$ and at most $\$10,000$ in corporate bonds yielding $3\%$.

a. Using $x$ to denote the amount of money invested in Treasury bills and y to denote the amount invested in corporate bonds, write a system of linear inequalities that describes the possible amounts of each investment.

b. Graph the system and label the corner points.

To determine

To find:

1. To write a system of linear inequalities that describes the possible amounts of each investment.

2. To graph the system and label the corner points.

Answer to Problem 57AYU

Solution:

$x \ge 0, y \ge 0, x + y \le 50,000, x \ge 35000, y \le 10000$

Explanation of Solution

Given:

• Total investment upto $\$50,000$

• Minimum investment in Treasury Bills $\$35,000$

• Maximum investment in corporate bond $\$10,000$

• Interest from Treasury $\text{Bills} = 1\%$

• Interest from junk $\text{bond} = 3\text{\%}$

Calculation:

1. Begin by assigning symbols for the two variables.

$x = \text{Amount invested in Treasury Bills}.$

$y = \text{Amount invested in corporate bond}.$

Because $x$ and $y$ represents the amount invested, the only meaningful values of $x$ and $y$ are non–negative.

Therefore, $x \ge 0, y \ge 0$

From the given data we get

$x + y \le 50,000$

$x \ge 35,000$

$y \le 10,000$

Therefore, a system of linear inequalities that describes the possible amounts of each investment are

$x \ge 0$

$y \ge 0$

$x + y \le 50,000$

$x \ge 35,000$

$y \le 10,000$

2. The graph of the system and corner points are illustrated in the figure below.