A financier plans to invest up to $500,000 in two projects. Project A yields a return of 8% on the investment of x dollars, whereas Project B yields a return of 13% on the investment of y dollars. Because the investment in Project B is riskier than the investment in Project A, she has decided that the investment in Project B should not exceed 40% of the total investment. How much should the financier invest in each project in order to maximize the return on her investment?
(x, y) =
What is the maximum return?
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See your previous post.
To find the amount the financier should invest in each project (x, y), we need to maximize the return on her investment. Let's break down the problem step by step:
Step 1: Set up the constraints
- The total investment amount is $500,000, so we have the constraint x + y = $500,000.
- The investment in Project B should not exceed 40% of the total investment, so we have the constraint y ≤ 0.4 * $500,000.
Step 2: Determine the return on investment for each project
Project A yields a return of 8% on the investment of x dollars, so the return from Project A would be 0.08x.
Project B yields a return of 13% on the investment of y dollars, so the return from Project B would be 0.13y.
Step 3: Define the objective function
The objective is to maximize the return on the investment. So, we want to maximize the total return, which is the sum of the returns from Project A and Project B. Let's call it R:
R = 0.08x + 0.13y
Step 4: Solve the optimization problem
Using the constraints and the objective function, we can now solve the optimization problem. We can do this by substituting the value of y from the first constraint into the objective function, i.e., substitute y = $500,000 - x:
R = 0.08x + 0.13($500,000 - x)
Now, we have an expression for the total return R in terms of x only. We can simplify it further:
R = 0.08x + 0.13($500,000) - 0.13x
R = 0.08x + $65,000 - 0.13x
R = -0.05x + $65,000
Step 5: Maximize the return
To maximize the return, we need to find the value of x that maximizes the objective function R. In this case, since the coefficient of x is negative (-0.05x), the maximum value will occur when x is at its lower limit, i.e., x = 0.
Step 6: Calculate the value of y
Substituting x = 0 back into the first constraint, we find y = $500,000 - 0 = $500,000.
So, the maximum return is obtained when the financier invests $0 in Project A and $500,000 in Project B.
Hence, (x, y) = ($0, $500,000) and the maximum return is $65,000.