The Fiedler family has up to $130,000 to invest. They decide that they want to have at least $40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be invested in more volatile bonds yielding 11%. How much should they invest in each type of bond to maximize income if the amount in the stable bond should not exceed the amount in the more volatile bond? What is the maximum income?
To solve this problem and find the optimal investment strategy, we can use a mathematical technique called linear programming. Linear programming helps us determine how to maximize or minimize a given objective (in this case, income) while considering certain constraints or limitations.
Let's denote the amount invested in stable bonds as x (in dollars) and the amount invested in more volatile bonds as y (also in dollars). According to the problem, we have the following constraints:
1. The total investment should not exceed $130,000: x + y ≤ 130,000
2. At least $40,000 should be invested in stable bonds: x ≥ 40,000
3. No more than $60,000 should be invested in volatile bonds: y ≤ 60,000
4. The amount in stable bonds should not exceed the amount in volatile bonds: x ≤ y
Now, let's define the objective function, which is the income generated by these investments. The income from the stable bonds can be calculated by multiplying the amount invested in stable bonds (x) by the interest rate (5.5% or 0.055). Similarly, the income from the volatile bonds can be calculated by multiplying the amount invested in volatile bonds (y) by the interest rate (11% or 0.11).
The objective function for maximizing income is:
Income = 0.055x + 0.11y
Now, to find the optimal investment strategy, we will solve this linear programming problem. This can be done using various methods, such as graphical analysis or a solver in spreadsheet software. Here, we will use the graphical method.
To construct the graph, we need to plot the feasible region defined by the constraints (1-4) and then evaluate the objective function at each corner point of the feasible region to find the maximum income.
By solving the system of equations formed by the constraints and analyzing the feasible region, we find that the corner points are:
Corner point A: x = 40,000, y = 60,000
Corner point B: x = 40,000, y = 90,000
Corner point C: x = 60,000, y = 60,000
Now, we substitute these corner points into the objective function and calculate the income:
At corner point A: Income = 0.055(40,000) + 0.11(60,000) = $9,300
At corner point B: Income = 0.055(40,000) + 0.11(90,000) = $12,350
At corner point C: Income = 0.055(60,000) + 0.11(60,000) = $9,900
From these calculations, we see that point B yields the maximum income of $12,350.
Therefore, the Fiedler family should invest $40,000 in stable bonds yielding 5.5% and $90,000 in more volatile bonds yielding 11% to maximize their income, which would be $12,350.