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?
P= 0.055x + 0.11y
x+y <= 130,000
x>= 40,000
y <= 60,000
Stable bond: 130.00 - 60,000 = 70,000
How do I solve this problem using linear programming?
With linear programming, you have to graph each of the equations.
x<=y also is needed along with your other equations.
Shade in the correct area for the inequalities and look at your intersection points.
linear programming graphically?
x axis is amount in stable bonds
y axis is amount in volitile bonds
ok, then mark two points on x axis, 40k,and 130k
then points on y axis 60k and 0k
So the chosen decision points on max and min are (draw horizontal lines lightly)
x=40, y=60, y=0 (min), x=130k max
now draw a line connecting (60,70) to (130,0)
cross hatch the area enclosed by (40,0)(40,60)(70,60)(130,0) That enclosed area is the area of possible investments of up to 130K dollars.
Now, there is a nice theorem that tells us the max and min profit is on those end points that enclose the area. Figure the profit at each corner, and you will have the max profit point for x,y
you did not follow one of the restraints.
your answer is wrong.
re-read the restraints.
To solve this problem using linear programming, you can follow these steps:
1. Define the decision variables:
- Let x represent the amount invested in stable bonds (in dollars).
- Let y represent the amount invested in volatile bonds (in dollars).
2. Write the objective function:
The objective is to maximize the income, which is given by the equation P = 0.055x + 0.11y.
3. Write the constraints:
a) The total amount invested cannot exceed $130,000, so the constraint is: x + y ≤ 130,000.
b) The amount invested in stable bonds should be at least $40,000, so the constraint is: x ≥ 40,000.
c) The amount invested in volatile bonds should be no more than $60,000, so the constraint is: y ≤ 60,000.
4. Graph the feasible region:
Plot the constraints on a graph to visualize the feasible region. This region represents the combinations of x and y values that satisfy all the constraints.
5. Identify the corner points of the feasible region:
Find the intersection points of the lines or boundaries determined by the constraints. These points will be the vertices of the feasible region.
6. Evaluate the objective function at the corner points:
Plug in the x and y values of each corner point into the objective function P = 0.055x + 0.11y and calculate the corresponding income.
7. Determine the optimal solution:
Compare the income values obtained at each corner point and identify the maximum income. The corresponding values of x and y will give the optimal solution.
In this specific case, the feasible region is a triangle since it is bound by the three constraints. You can analyze the corner points of the triangle to find the maximum income and the corresponding values of x and y that achieve it.