If the 3 amounts are x,y,z for 7%,8%,12% respectively, then we want to
maximize .07x + .08y + .12z subject to
x+y+z = 12000
x >= 3z
z <= 2000
use your favorite tool to calculate the result.
please help me find the equation, contraints and the answer.....
please and thank you..
maximize .07x + .08y + .12z subject to
x+y+z = 12000
x >= 3z
z <= 2000
use your favorite tool to calculate the result.
Let's define the variables:
x = amount invested in municipal bonds (in dollars)
y = amount invested in bank certificates (in dollars)
According to the problem, the amount in high-risk bonds will be 12,000 - x - y.
Objective Function:
To maximize the annual interest yield, we need to maximize the sum of the interest earned from each type of bond:
Interest = 0.07x + 0.08y + 0.12(12,000 - x - y)
Constraints:
1. The amount invested in municipal bonds should be at least three times the amount invested in bank certificates:
x >= 3y
2. The amount invested in high-risk bonds should be no more than $2000:
12,000 - x - y <= 2000
Now, let's solve the problem using these equations and constraints.
First, substitute the expression for high-risk bonds into the objective function:
Interest = 0.07x + 0.08y + 0.12(12,000 - x - y)
Interest = 0.07x + 0.08y + 1,440 - 0.12x - 0.12y
Interest = 1,440 - 0.05x - 0.04y
Next, we can graph the feasible region of the problem by plotting the constraints and shading the area that satisfies all of them. The area within the feasible region represents all possible combinations of x and y.
Once we have the feasible region graph, we can find the corner points inside the region. Evaluate the objective function at each corner point and identify the one that gives the maximum interest yield. This will be our optimal solution.
To determine the maximum possible interest yield when the maximum invested in high-risk bonds is $3000, we need to update the constraints.
New constraint (updated maximum investment):
12,000 - x - y <= 3000
Repeat the same steps as before to solve for the maximum interest yield with the new constraint. Compare it to the previous maximum interest yield to find the difference.
It is important to note that these calculations might require a graphing calculator, software, or an optimization tool to solve.
Step 1: Define the variables
Let x be the amount invested in municipal bonds.
Let y be the amount invested in bank certificates.
Then, the amount invested in high-risk bonds will be 12,000 - x - y.
Step 2: Write the equation for the total investment
The total investment is $12,000, so we have:
x + y + (12,000 - x - y) = 12,000
12,000 - x - y = 12,000
Step 3: Write the equation for the maximum investment in high-risk bonds
The maximum investment in high-risk bonds is $2,000, so we have:
12,000 - x - y ≤ 2,000
-x - y ≤ -10,000
Step 4: Write the equation for the minimum investment in municipal bonds
The amount invested in municipal bonds should be at least three times the amount invested in bank certificates, so we have:
x ≥ 3y
Step 5: Write the equation for the interest yield
The interest yield is given by the sum of the interest from each type of bond. Since the woman wants to maximize her annual interest yield, we want to maximize the following equation:
0.07x + 0.08y + 0.12(12,000 - x - y)
Step 6: Combine the equations and constraints
Combining all the equations and constraints, we have the following linear programming problem:
Maximize: 0.07x + 0.08y + 0.12(12,000 - x - y)
Subject to:
12,000 - x - y = 12,000
-x - y ≤ -10,000
x ≥ 3y
Now, to find the solution to this linear programming problem, we can use optimization techniques such as the Simplex Method or graphical methods.