# A pension fund manager decides to invest a total of at most $40 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 8% annual interest. He plans to invest at least...

A pension fund manager decides to invest a total of at most $40 million in U.S. Treasury bonds paying 6% annual interest and in mutual funds paying 8% annual interest. He plans to invest at least $5 million in bonds and at least $20 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 interest?

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Hello!

Such problems are called linear programming ones. We have a two-dimensional domain formed by straight lines, and a linear function to maximize on this domain. To solve such a problem it is desirable to draw the domain and a line where the function to maximize has some value (all such lines are parallel).

Denote a number of millions invested in bonds as `x` and in funds as `y.` Then the domain is formed by the inequalities

`x+ylt=40`

`xgt=5`

`ygt=20`

`100x+200ylt=7000,` or `x+2ylt=70.`

They form a quadrilateral domain, please look at the picture given by the attached link. The function to maximize is

`x*1,000,000*6%+y*1,000,000*8%=`

`=60,000x+80,000y=10,000(6x+8y).`

The line drawn in orange is `6x+8y=350.` The parallel line `6x+8y=A,` touching the domain and with the greatest `A` is the desired one.

It is evident from the picture that the point `(10,30)` is the nearest to the orange line. This means **$10 million** should be invested in U.S. Treasury bonds and **$30 million** in mutual funds. The annual interest will be `10,000(6*10+8*30)=` **3,000,000($)**.