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?
you can go through the calculations, but clearly the maximum income will happen if the full $60K is invested in the high-interest bonds, with an additional $60K in the low-yield instrument.
To solve this problem, we can use linear programming, which involves defining constraints and an objective to optimize. In this case, the constraints are the minimum and maximum investments in each type of bond, and the objective is to maximize income.
Let's define the variables:
Let x represent the amount invested in stable bonds yielding 5.5%.
Let y represent the amount invested in more volatile bonds yielding 11%.
Now, let's set up the constraints:
1. The total amount invested cannot 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 more volatile bonds:
y ≤ 60,000
4. The amount invested in stable bonds should not exceed the amount invested in more volatile bonds:
x ≤ y
Now, let's define the objective function, which is the income:
Income = (x * 0.055) + (y * 0.11)
To maximize the income, we need to solve this linear programming problem.
We can use a graphing tool or linear programming software to find the solution, or we can use a graphical method to visualize the feasible region and find the optimal solution.
Graphing the constraints, we can plot the lines x + y = 130,000, x = 40,000, y = 60,000, and x = y. By shading the region that satisfies all the constraints, we can identify the feasible region.
Once we have the feasible region, we need to check the corner points (vertices) of the feasible region to find the maximum income.
The corner points of the feasible region are:
A: (40,000, 40,000)
B: (60,000, 60,000)
C: (40,000, 60,000)
D: (60,000, 60,000)
Now, we substitute these corner points into the income function and calculate the income:
Income at point A = (40,000 * 0.055) + (40,000 * 0.11) = $6,600
Income at point B = (60,000 * 0.055) + (60,000 * 0.11) = $11,100
Income at point C = (40,000 * 0.055) + (60,000 * 0.11) = $9,300
Income at point D = (60,000 * 0.055) + (60,000 * 0.11) = $11,100
From the calculations above, we can see that the maximum income is $11,100, which occurs at both points B and D. Therefore, the maximum income is $11,100.
To answer the second part of the question, the amount to be invested in each type of bond to achieve the maximum income is:
x = y = 60,000
So, the Fiedler family should invest $60,000 in stable bonds yielding 5.5% and $60,000 in more volatile bonds yielding 11% to maximize their income, achieving a maximum income of $11,100.