A couple wants to invest up to
$
60000
$60000. They can purchase a type A bond yielding
9.25
%
9.25% return and a type B bond yielding a
12.5
%
12.5% return on the amount invested. They also want to invest at least as much in the type A bond as in the type B bond. They will also invest at least
$
30000
$30000 in type A and no more than
$
36000
$36000 in type
B
B bond.
How much should they invest in each type of bond to maximize their return?
Let
x
x be the amount invested in type
A
A bond and
y
y the amount invested in type
B
B bond. a) Write the objective function to be maximized. b) The feasible region has 3 corner points. List the three corner points. c) Determine the amount to be invested in each bond to maximize the return and determine the maximum return: Amount to be invested in type
A bonds \$
Amount to be invested in type
B bonds \$ Maximum return is
a) The objective function to be maximized is the total return on the investment.
b) The feasible region has 3 corner points:
Corner Point 1: (x, y) = ($30,000, $30,000)
Corner Point 2: (x, y) = ($36,000, $36,000)
Corner Point 3: (x, y) = ($36,000, $30,000)
c) We need to determine the amount to be invested in each bond to maximize the return and find the maximum return.
To simplify the calculations, let's assume the number of dollars invested in type A bond is denoted by "a" and the number of dollars invested in type B bond is denoted by "b".
Since they want to invest at least as much in type A bond as in type B bond, we have the constraint:
a >= b
The total investment should not exceed $60,000, so we have the constraint:
a + b <= $60,000
They need to invest at least $30,000 in type A bond, so we have the constraint:
a >= $30,000
They cannot invest more than $36,000 in type B bond, so we have the constraint:
b <= $36,000
We can represent the feasible region graphically using these constraints. The feasible region is the shaded region where the constraints are satisfied. The three corner points mentioned earlier (from left to right) correspond to the intersections of the lines and curves on the graph.
Now, we need to calculate the return on investment at each of the corner points.
Corner Point 1: (x, y) = ($30,000, $30,000)
Return on Investment = (0.0925 * $30,000) + (0.125 * $30,000) = $2,775 + $3,750 = $6,525
Corner Point 2: (x, y) = ($36,000, $36,000)
Return on Investment = (0.0925 * $36,000) + (0.125 * $36,000) = $3,330 + $4,500 = $7,830
Corner Point 3: (x, y) = ($36,000, $30,000)
Return on Investment = (0.0925 * $36,000) + (0.125 * $30,000) = $3,330 + $3,750 = $7,080
Therefore, the maximum return is $7,830, and it can be achieved by investing $36,000 in type A bonds and $36,000 in type B bonds.
a) The objective function to be maximized is the return on investment. Let R represent the return on investment.
b) The feasible region has 3 corner points. The corner points are:
- (30000, 30000)
- (36000, 30000)
- (36000, 36000)
c) To determine the amount to be invested in each bond to maximize the return, we can set up a linear programming problem.
Let x be the amount invested in type A bond and y be the amount invested in type B bond.
The constraints are:
- 30000 ≤ x ≤ 36000
- 0 ≤ y ≤ 60000
- x ≥ y
The objective is to maximize the return, which can be calculated as:
R = 0.0925x + 0.125y
To solve this linear programming problem, we can graph the feasible region and find the corner point that results in the maximum return.
The three corner points are:
- (30000, 30000)
- (36000, 30000)
- (36000, 36000)
By substituting the values of the corner points into the objective function, we can calculate the returns:
- R1 = 0.0925(30000) + 0.125(30000)
- R2 = 0.0925(36000) + 0.125(30000)
- R3 = 0.0925(36000) + 0.125(36000)
By comparing the returns, we can determine the maximum return and the corresponding investment amounts.
The amounts to be invested in each bond to maximize the return are:
- Amount to be invested in type A bonds: $36000
- Amount to be invested in type B bonds: $30000
The maximum return is:
- Maximum return is R3 = 0.0925(36000) + 0.125(36000) = $9270
a) The objective function to be maximized is the total return on the investment. Let R represent the total return.
b) To find the corner points of the feasible region, we need to consider the given constraints. The constraints are:
1. The total investment cannot exceed $60000.
x + y ≤ 60000
2. The investment in type A bond should be at least as much as in type B bond.
x ≥ y
3. The investment in type A bond should be at least $30000.
x ≥ 30000
4. The investment in type B bond should be no more than $36000.
y ≤ 36000
By graphing these constraints, we can determine the three corner points of the feasible region:
Point 1: (30000, 30000)
Point 2: (36000, 0)
Point 3: (48000, 12000)
c) To determine the amount to be invested in each bond to maximize the return, we need to evaluate the objective function at each corner point.
1. For Point 1: (30000, 30000)
Total return = 0.0925x + 0.125y
= 0.0925(30000) + 0.125(30000)
= 2775 + 3750
= 6525
2. For Point 2: (36000, 0)
Total return = 0.0925x + 0.125y
= 0.0925(36000) + 0.125(0)
= 3330 + 0
= 3330
3. For Point 3: (48000, 12000)
Total return = 0.0925x + 0.125y
= 0.0925(48000) + 0.125(12000)
= 4440 + 1500
= 5940
To maximize the return, the couple should invest $30000 in type A bonds and $30000 in type B bonds, resulting in a maximum return of $6525.