Apension fund manager decides to invest a total of at most $3535 million in u. s. treasury bonds paying 66% annual interest and in mutual funds paying 99% annual interest. he plans to invest at least $55 million in bonds and at least $1515 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 $66000 on fees. how much should be invested in each to maximize annual interest? what is the maximum annual interest?

The "maximum annual interest" is "1536.15 million dollars".

Step-by-step explanation:

Let "t" represent the "money" (in millions) invested in "US Treasury bonds" and "f" the "money" (in millions) invested in "mutual funds".

t ≥ 0

f ≥ 0

A "pension fund manager" decided to "invest a total" of at most "$ 3535 million" in "US Treasury bonds" paying "66% annual interest" and in "mutual funds" paying "99% annual interest".

t + f ≤ 40

t ≥ 5

f≥ 15

Bonds which have an "initial fee of $100 per million dollars", while "the fee for mutual funds" is "$200 per million". The "fund manager" is permitted to spend no more than "$66000 on fees".

100t + 200f ≤ 66000 divide both sides by 100

t + 2f ≤ 660

The annual interest is described with the objective function:

F (t, f) = 0.66t + 0.99f

We have the following constrains:

t ≥55

f ≥ 1515

t + f ≤3535

t + f ≤ 660

The corner points are: (55, 1515)

Evaluating the function at the corner points, we find:

F(55, 1515) = 0.66 × 55 + 0.99 × 1515 = 136.3 + 1499.85 = 1536.15

The "objective function", represents "revenue", is maximized when "t = 55" and "f = 1515 ".

The manager should invest 55 million in US Treasury bond and 1515 millions in mutual funds.

The "maximum annual interest" is "1536.15 million dollars".

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