Chapter 3.8, Problem 4P

Explanation of Solution

 Formulation of a Linear Programming (LP) to maximize total profit earned from Erica’s investment:

 Let “x₁” be the number of dollars invested in stocks.

 The “x₂” be the number of dollars invested in loans.

 As every dollar invested in stocks gives 10 cents profits, so the total profit on stocks would be $0.1x₁. Likewise, the total profit on loan would be $0.15x₂.

 Since the objective is to maximize total profit earned from Erica’s investment, the objective function is defined as follows:

 $\text{Max Z}=0.10x_1+0.15x_2$

 Constraint 1:

 At least, 30% of total invested money must be invested in stocks. The “x₁+ x₂” is the total money invested in the investment. Since the number of dollars invested in stocks must be at least 30% of the total investment, the constraint is defined as follows:

 $\begin{array}{l}{x}_1\ge 0.3\left(x_1+x_2\right)\\ 0.7x_1-0.3x_2\ge 0\end{array}$

 Constraint 2:

 At least, $400 of all money must be invested in loans, so the constraint is defined as:

 $x_2\ge 400$

 Constraint 3:

 At most, $1000 is invested in stocks and loans, so the constraint is defined as:

 $x_1+x_2\le 1000$

 Therefore, the formulation to the given linear program is:

 $\text{Max Z}=0$