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?
first, put the words into algebra. If there is x at 5.5% and y at 11%, you have:
x+y <= 130000
x >= 40000
y <= 60000
maximize p = .055x + .11y
Now just use your favorite linear programming tool.
To solve this problem, we can use a linear programming approach. Let's define the following variables:
Let x be the amount invested in stable bonds yielding 5.5%.
Let y be the amount invested in volatile bonds yielding 11%.
We need to set up the constraints based on the given conditions:
1. The total investment should not exceed $130,000:
x + y ≤ 130,000
2. The amount invested in stable bonds should be at least $40,000:
x ≥ 40,000
3. The amount invested in volatile bonds should not exceed $60,000:
y ≤ 60,000
4. The amount invested in stable bonds should not exceed the amount invested in volatile bonds:
x ≤ y
Since we want to maximize the income, we can define the objective function:
Income = 0.055x + 0.11y
Now, let's graph these constraints on a coordinate plane:
First, plot the line x = 40,000 (vertical line at x = 40,000).
Next, plot the line y = 60,000 (horizontal line at y = 60,000).
Then, plot the line x = y (diagonal line with a slope of 1 passing through the origin).
Finally, plot the line x + y = 130,000 (line with a slope of -1 passing through the point (130,000, 0)).
The feasible region is the region where all the constraints intersect. It should be a triangle-shaped region.
To find the vertices of this feasible region, we can solve the system of equations formed by the lines:
Solving x = 40,000 and x = y, we get:
x = 40,000
y = 40,000
Solving y = 60,000 and x = y, we get:
x = 60,000
y = 60,000
Solving x = y and x + y = 130,000, we get:
x = 65,000
y = 65,000
The feasible region has vertices at (40,000, 40,000), (60,000, 60,000), and (65,000, 65,000).
Now, evaluate the objective function at each vertex to find the maximum income:
Income(40,000, 40,000) = 0.055 * 40,000 + 0.11 * 40,000 = $4,400
Income(60,000, 60,000) = 0.055 * 60,000 + 0.11 * 60,000 = $6,600
Income(65,000, 65,000) = 0.055 * 65,000 + 0.11 * 65,000 = $7,150
Therefore, the maximum income the Fiedler family can earn is $7,150, and it occurs when they invest $65,000 in both stable and volatile bonds.