A woman wishes to invest $12,000 in three of bonds: municipal bonds paying 7% interest per year, bank investment certificates paying 8%, and high-risk bonds paying 12%. For tax reasons she wants the amount invested in municipal bonds to be at least three times the amount invested in bank certificates. To keep her level of risk manageable, she will invest no more than $2000 in high risk bonds. How much should she invest in each type of bond to maximize her annual interest yield? [Hint: Let x = amount in municipal bonds and y = amount in bank certificates. Then the amount in high-risk bonds will be 12,000 – x – y]. Hence, suppose the investor decides to increase the maximum invested in high-risk bonds to $3000 but leaves the other conditions unchanged. By how much will her maximum possible interest yield increase?

please help me find the equation, contraints and the answer.....
please and thank you..

To maximize the annual interest yield, we need to create an objective function and set up constraints.

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.

To find the equation and constraints for this problem, let's break it down step-by-step:

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.

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.