SET UP BUT DO NOT SOLVE THE FOLLOWING LINEAR PROGRAMMING PROBLEM.

A financier plans to invest up to $500,000 in two products. Project A yields a return of 10% on an 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 in order to maximize her return? What is the maximum return?

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I'm pretty sure that the profit function is P=0.1A+0.15B

And part of the "subject to" is A+B≤500,000
A≥0, B≥0

But I can't figure out how to apply the part in which "Project B should not exceed 40% of the total investment."

40% of total investment

= 40%*500000
= 200000
This puts a ceiling on B, and a lower bound on A.

To incorporate the constraint that the investment in Project B should not exceed 40% of the total investment, you need to introduce an additional constraint.

Let's assume that the total investment is denoted as T. Then, the constraint can be written as B ≤ 0.4T.

Since we know that the total investment T is $500,000, we can substitute this value into the constraint to get B ≤ 0.4(500,000).

The complete set of constraints for this problem is as follows:

1. A + B ≤ 500,000 (total investment constraint)
2. B ≤ 0.4(500,000) (constraint for maximum investment in Project B)
3. A ≥ 0 (non-negativity constraint for Project A)
4. B ≥ 0 (non-negativity constraint for Project B)

Together with the profit function, P = 0.1A + 0.15B, these constraints form the linear programming problem.