A financier plans to invest up to $2 million in three projects. She estimates
that Project A will yield a return of 10% on her investment, Project B will yield a return of 15% on her
investment, and Project C will yield a 20% return on her investment. Because of the risks associated with
the investments, she decided to put not more than 20% of her total investment in Project C. She also decided
that her investments in Projects B and C should not exceed 60% of her total investment. Finally, she decided
that her investment in Project A should be at least 60% of her investments in Projects B and C. How much
should the financier invest in each project if she wishes to maximize the total returns on her investments? set up but do not solve!
To solve this problem, we need to define some variables and set up a system of equations based on the given conditions.
Let:
- x be the amount invested in Project A
- y be the amount invested in Project B
- z be the amount invested in Project C
Based on the given conditions, we can set up the following equations:
1. The total investment constraint:
x + y + z ≤ 2,000,000 (investing up to $2 million)
2. Project C's investment constraint:
z ≤ 0.2(x + y + z) (investing not more than 20% in Project C)
3. Projects B and C's investment constraint:
y + z ≤ 0.6(x + y + z) (investing not more than 60% in Projects B and C)
4. Project A's investment constraint:
x ≥ 0.6(y + z) (investing at least 60% in Project A compared to Projects B and C)
These equations represent the given conditions for the investments. By solving this system of equations, we can find the optimal investment amounts for each project to maximize the total returns.