Clayton is planning to invest up to 40,000 in corporate and...

Clayton is planning to invest up to $\$ 40,000$ in corporate and municipal bonds. The least he is allowed to invest in corporate bonds is $\$ 6000,$ and he does not want to invest more than $\$ 22,000$ in corporate bonds. He also does not want to invest more than $\$ 30,000$ in municipal bonds. The interest is $8 \%$ on corporate bonds and $7 \frac{1}{2} \%$ on municipal bonds. This is simple interest for one year. How much should he invest in each type of bond in order to maximize his income? What is the maximum income?

Key Concepts

Linear Programming

Linear programming is a mathematical technique used to optimize a linear objective function, subject to a set of linear constraints. It is widely applied in various fields to determine the best achievable outcome when there are requirements or limitations represented by linear relationships.

Objective Function

The objective function defines what needs to be maximized or minimized in a linear programming problem. In investment problems, it typically represents the total return or income, expressed as a linear combination of investments in different options, each multiplied by their respective interest rates or returns.

Constraints

Constraints are the limitations or conditions imposed on the decision variables in a linear programming problem. They are expressed as linear inequalities or equations, such as investment limits or minimum and maximum allowable amounts, which define the boundaries that any solution must satisfy.

Feasible Region

The feasible region is the set of all possible solutions that satisfy the constraints of a linear programming problem. In graphical methods, this region is often represented by the intersection of half-planes, and the optimal solution is found at one of the vertices, according to the corner point theorem.

Simple Interest

Simple interest refers to interest calculated only on the principal amount of an investment, rather than on accrued interest. This linear relationship between the principal, rate, and time makes it suitable for linear programming formulations, as the return can be expressed as a straightforward linear function of the investment amount.

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