How would you set this one up as well?
5. You have up to $30,000 to invest. Your broker recommends investing in AAA bonds yielding 8% and B+ bonds yielding 12%. You decide to invest at most $12,000 in the B+-rated bond and at least $6,000 in the AAA bond. The amount invested in the AAA bond must exceed or equal the amount in the B+ bond. How much should be invested in each to maximize the return on the investment?
A + B = < 30,000
A > = B
B = < 12,000
B return > A return
In order to maximize money we have to invest it. The rate of retern is highest with the B bond
so, A + 12,000 = 30,000
B = 12,000
A = 18,000
Thanks for all your help on both problems!! :)
To solve this problem, we need to find the amount to be invested in AAA bonds and B+ bonds that will maximize the return on investment.
Let's represent the amount invested in AAA bonds as 'x' and the amount invested in B+ bonds as 'y'.
Based on the given information, we have the following conditions:
1. The total amount invested should not exceed $30,000:
x + y ≤ $30,000
2. The amount invested in B+ bonds should be at most $12,000:
y ≤ $12,000
3. The amount invested in AAA bonds should be at least $6,000:
x ≥ $6,000
4. The amount invested in AAA bonds should be greater than or equal to the amount invested in B+ bonds:
x ≥ y
Now, let's proceed with finding the optimal investment strategy.
To maximize the return on investment, we need to maximize the interest earned. The interest earned from AAA bonds would be 8% of the investment amount, and from B+ bonds would be 12% of the investment amount.
Let's denote the interest earned from AAA bonds as 'i_aaa' and from B+ bonds as 'i_b+':
i_aaa = 0.08x
i_b+ = 0.12y
We want to maximize the total interest earned, so our objective function will be:
Total interest = i_aaa + i_b+
= 0.08x + 0.12y
Now, we have our objective function and the set of constraints. We can solve this problem using linear programming techniques or graphically by plotting the feasible region.
However, since the feasible region is relatively simple, we can directly substitute the different values of x and y to find the maximum total interest.
Let's consider three cases:
Case 1: x = $6,000 and y = $12,000
Total interest = 0.08($6,000) + 0.12($12,000) = $2,280
Case 2: x = $12,000 and y = $12,000
Total interest = 0.08($12,000) + 0.12($12,000) = $2,400
Case 3: x = $18,000 and y = $12,000
Total interest = 0.08($18,000) + 0.12($12,000) = $2,520
Comparing these three cases, we can observe that the highest total interest is obtained when x = $18,000 and y = $12,000. Therefore, to maximize the return on investment, $18,000 should be invested in AAA bonds and $12,000 should be invested in B+ bonds.