A financier plans to invest up to $500,000 in two projects. Project A yields a return of 11% on the investment of x dollars, whereas Project B yields a return of 16% 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?

Ah, the world of finance, where numbers dance and investors juggle their dollars. Let's see how we can help this financier make some wise decisions.

Let's start by assigning variables to our unknowns. Let x represent the amount invested in Project A and y represent the amount invested in Project B. Since the total investment is limited to $500,000, we know that x + y ≤ 500,000.

Now, we move on to the returns on investment. Project A yields a return of 11% on the investment of x dollars, while Project B yields a return of 16% on the investment of y dollars. To maximize the return on her investment, the savvy financier would want to maximize the sum of these returns.

However, our generous financier also keeps an eye on risk. She has smartly decided that the investment in Project B should not exceed 40% of the total investment. In other words, y ≤ 0.4 * (x + y), which simplifies to y ≤ 0.4x + 0.4y.

Now, let's put all these pieces of the puzzle together. We want to maximize the sum of the returns, which can be calculated as 0.11x + 0.16y. At the same time, we need to satisfy the constraints x + y ≤ 500,000 and y ≤ 0.4x + 0.4y. Time to put on our thinking caps!

But wait, let's not get too serious! Investing can be fun too. Picture a clown juggling your dollars...or maybe not, we don't want any financial disasters. So let's calculate the optimal investment amounts and maximize those returns!

To maximize the return on investment, the financier should invest in Project A and Project B accordingly. Let's start by setting up the equations.

Let x represent the amount invested in Project A (in dollars).
Let y represent the amount invested in Project B (in dollars).

The total investment should not exceed $500,000:
x + y ≤ 500,000

The return on investment for Project A is 11%:
Return from Project A = 0.11x

The return on investment for Project B is 16%:
Return from Project B = 0.16y

Since the investment in Project B should not exceed 40% of the total investment, we have:
y ≤ 0.4 * (x + y)

We want to maximize the return, which can be represented as follows:
Maximize: Return from Project A + Return from Project B

To solve this problem, we can use linear programming.

To maximize the return on investment, the financier needs to find the optimal allocation of funds between Project A and Project B. Let's start by defining the investment amounts for each project.

Let:
- x be the amount invested in Project A (in dollars)
- y be the amount invested in Project B (in dollars)

From the given information, we know that:
- The total investment is limited to $500,000, so x + y ≤ 500,000.
- The return on investment for Project A is 11%, so the return from Project A would be 0.11x.
- The return on investment for Project B is 16%, so the return from Project B would be 0.16y.
- Project B cannot exceed 40% of the total investment, so y ≤ 0.4(x + y).

To maximize the return, we need to set up an objective function. Since we want to maximize the return on investment, we can define our objective function as the sum of the returns from both projects:

f(x, y) = 0.11x + 0.16y

Now, let's set up the optimization problem. We want to maximize f(x, y) subject to the constraints mentioned earlier:

Maximize: f(x, y) = 0.11x + 0.16y
Subject to:
- x + y ≤ 500,000
- y ≤ 0.4(x + y)

To solve this optimization problem, we can use various methods such as graphical analysis, linear programming, or calculus. Let's use the graphical method:

1. Draw the feasible region by graphing the constraint inequalities. In this case, we have a linear feasible region.
- Draw the line x + y = 500,000.
- Draw the line y = 0.4(x + y).

2. Shade the region that satisfies all constraints. This shaded region represents the feasible region.

3. Evaluate the objective function f(x, y) at the corner points of the feasible region.

4. Select the corner point that maximizes the objective function. This will be the optimal solution.

Based on the constraints and objective function, this graphical method will help us find the optimal allocation of funds between Project A and Project B that maximizes the return on investment.

You want to maximize

p = 0.11x + 0.16y subject to
x+y = 500000
y <= 0.40(x+y)

So, graph the region and pick the vertex which maximizes the income.

I get (in 10,000's) p(30,20) = 6.50