A pension 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?

Answer: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".