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?
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To solve this problem, we need to set up an optimization problem. 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 make sure that the total investment is not more than $130,000:
x + y ≤ 130,000
We also have the following constraints:
x ≥ 40,000 (to have at least $40,000 invested in stable bonds)
y ≤ 60,000 (to have no more than $60,000 invested in volatile bonds)
The objective is to maximize the income. The income from the stable bonds yielding 5.5% will be 0.055x, and the income from the volatile bonds yielding 11% will be 0.11y. So, the total income, denoted as I, will be:
I = 0.055x + 0.11y
To find the optimal solution, we need to solve this optimization problem. We can use the simplex method or graphical method to solve it. However, let's use the graphical method to find the feasible region and the corner points.
Start by graphing the constraints:
1. Graph the line x + y = 130,000 (the total investment constraint).
2. Draw a dashed line for x = 40,000 (the constraint for the stable bonds).
3. Draw a horizontal line for y = 60,000 (the constraint for the volatile bonds).
The feasible region is the area where all the constraints are satisfied. It will be a triangle formed by the intersection of the three lines.
Now, we need to evaluate the objective function I = 0.055x + 0.11y for each corner point of the feasible region to find the maximum income.
The corner points of the feasible region are the vertices of the triangle. Find the coordinates of these points by solving the equations of the lines intersecting at those points.
Once you have the coordinates of the corner points, substitute the x and y values into the objective function I = 0.055x + 0.11y for each point to find the maximum income.
Compare the income values for each corner point to find the maximum income.
Once you find the maximum income, you can determine the allocation of the investment (x and y values) that yields the maximum income.