A financier plans to invest up to $450,000 in two projects. Project A yields a return of 10% on the investment, whereas Project B yields a return of 15% on the investment. Because the investment in Project B is riskier than the investment in Project A, the financier has decided that the investment in Project B should not exceed 40% of the total investment. How much should she invest in each project to maximize the return on her investment?
Project A ______ $
Project B ______ $
What is the maximum return?
$ _______
how did you get 0.6?
nevermind, i figured it out. thank you!
You're welcome.
To solve this problem, we can use a mathematical approach called linear programming. Linear programming is a method to find the maximum or minimum value of a linear objective function, subject to a set of linear inequality constraints.
Let's define the variables:
Let x represent the amount invested in Project A (in dollars)
Let y represent the amount invested in Project B (in dollars)
According to the given information, we have two constraints:
1. The total investment should not exceed $450,000.
This can be written as: x + y ≤ 450,000
2. The investment in Project B should not exceed 40% of the total investment.
This can be written as: y ≤ 0.4 * (x + y)
Simplifying, we have: 0.6y ≤ 0.4x
The objective is to maximize the return on investment (ROI), which is the total return from each project:
ROI = 0.1x + 0.15y
Now let's solve this linear programming problem graphically:
1. Plot the constraints on a graph:
- The first constraint, x + y ≤ 450,000, represents a straight line with a slope of -1 passing through the point (450,000, 0).
- The second constraint, 0.6y ≤ 0.4x, can be transformed to y ≤ 0.67x and represents a straight line with a slope of 0.67 passing through the origin (0, 0).
2. Find the feasible region:
The feasible region is the area where the constraints are satisfied. In this case, it is the region under the line x + y ≤ 450,000 and above the line y ≤ 0.67x.
3. Identify the corner points of the feasible region:
The corner points are the vertices of the feasible region. We can find them by solving the system of equations formed by the boundary lines.
By evaluating the constraints at the corner points, we can determine the maximum ROI.
To calculate the exact values, we need to solve the system of equations formed by the boundary lines:
Let's solve the system of equations:
1. y = 0.67x
2. x + y = 450,000
Substitute the first equation into the second equation:
x + 0.67x = 450,000
1.67x = 450,000
x = 450,000 / 1.67
x ≈ 269,461.08
Substitute x back into the first equation:
y = 0.67(269,461.08)
y ≈ 180,269.95
The corner point (269,461.08, 180,269.95) represents the maximum ROI.
Now let's calculate the maximum return on investment (ROI):
ROI = 0.1x + 0.15y
= 0.1(269,461.08) + 0.15(180,269.95)
≈ $44,846.11
Therefore, the maximum return on investment is approximately $44,846.11.
Now, let's find out how much should be invested in each project:
x = $269,461.08 (Project A)
y = $180,269.95 (Project B)
So, the amounts that should be invested in each project to maximize the return on the investment are:
Project A: $269,461.08
Project B: $180,269.95
0.6 * 450,000 = 270,000 > A
270,000 * 0.1 = 27,000 return