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?
$
Project A: 0.6*500,000 = $300,000 @ 8%.
Project B: 0.4*500,000 = $200,000 @ 13%.
(x, y) = (300000,200000).
I = 0.08*300,000 + 0.13*200,000 =
In order to maximize the return on her investment, the financier needs to determine the optimal investment amounts for Projects A and B.
Let's assume the financier invests a certain amount, x, in Project A. Then, the investment in Project B would be (500,000 - x), since the total investment is $500,000.
According to the problem statement, Project A yields a return of 8% on the investment of x dollars, while Project B yields a return of 13% on the investment of (500,000 - x) dollars.
The objective is to find the optimal values of x and (500,000 - x) that maximize the total return.
To do this, we need to set up an equation and then maximize it using calculus.
The total return (R) is given by:
R = 0.08x + 0.13(500,000 - x)
Since the investment in Project B should not exceed 40% of the total investment, we have the constraint:
x ≤ 0.6(500,000)
Simplifying the constraint, we have:
x ≤ 300,000
Now, we can maximize the total return equation (R) subject to the constraint.
Differentiating R with respect to x:
R' = 0.08 - 0.13
Setting R' equal to zero, we get:
0.08 - 0.13 = 0
Solving for x:
0.08x - 0.13(500,000 - x) = 0
0.08x - 0.13 * 500,000 + 0.13x = 0
0.21x = 0.13 * 500,000
x = (0.13 * 500,000) / 0.21
x = 100,000
Now we have the value of x, which is 100,000. The investment in Project A should be $100,000 in order to maximize the return on the investment.
To find the investment in Project B, we subtract x from the total investment:
Investment in B = (500,000 - x) = 500,000 - 100,000 = $400,000.
Therefore, the financier should invest $100,000 in Project A and $400,000 in Project B to maximize the return on her investment.
The maximum return can now be calculated using the values of x and y we found:
Maximum return = 0.08 * x + 0.13 * (500,000 - x)
= 0.08 * 100,000 + 0.13 * (500,000 - 100,000)
= 8,000 + 0.13 * 400,000
= 8,000 + 52,000
= $60,000
Therefore, the maximum return on the investment is $60,000.