Problema Solution
a pension fund manager decides to invest a total of at most $40 million in US Treasury bonds paying 6% annual interest and in mutual funds paying 9% annual interest. He plans to invest at least $5 million in bonds and at least $15 million in mutual funds. Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $7000 on fees. How much should be invested in each to maximize annual interest? what is the maximum annual interest?
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Let t represent the money (in millions) invested in US Treasury bonds and f the the money (in millions) invested in mutual funds
t >= 0
f >= 0
a pension fund manager decides to invest a total of at most $40 million in US Treasury bonds paying 6% annual interest and in mutual funds paying 9% annual interest
t + f <= 40
t >= 5
f >= 15
Bonds have an initial fee of $100 per million dollars, while the fee for mutual funds is $200 per million. The fund manager is allowed to spend no more than $7000 on fees.
100t + 200f <= 7000 divide both sides by 100
t + f <= 70
the annual interest is described with the objective function:
F(t, f) = 0.06t + 0.09f
We have the following constrains:
t >= 5
f >= 15
t + f <= 40
t + f < =70
click here to see the feasible region:
The feasible region is bounded so that a maximum exists.
The corner points are: (15, 5), (35, 5) and (15,25).
Evaluating the function at the corner points, we find:
F(15, 5) = 0.06*15 + 0.09*5 = 1.35
F(35, 5) = 0.06*35 + 0.09*5 = 2.55
F(15, 25) = 0.06*15 + 0.09*25 = 3.15
The objective function, which represents revenue, is maximized when t = 15 and f = 25.
The manager should invest 15 millions in US Treasury bond and 25 millions in mutual funds.
The maximum annual interest is 3.15 million dollars.