A pension fund manager decides to invest a total of at most $35 million in U.S. Treasury bonds paying 5% 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 $35 million in U.S. Treasury bonds paying 5% annual interest and in mutual funds paying 8% 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 alllowed to spend no more than $6000 on fees. How much should be invested in each to maximize annual interest? 

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lemjay | High School Teacher | (Level 3) Senior Educator

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Let x be the amount in millions dollar invested in bonds paying 5% annual interest. And let y be the amount in million dollars invested in mutual funds paying 8% annual interest.

Then, determine the constraints for each given conditions.

For the first condtion, the manager decides to invest a total of at most $35 millions. So, its equivalent equation is:

`x + y lt= 35`

Next, he plans to invest at least $5 million in bonds and at least $15 millions in mutual funds. So, the equivalent equations are:

`x gt= 5`

`ygt=15`

Also, the manager is allowed to spend no more that $6000 on fees. Since the initial fees for bonds is $100 per million dollars and for mutual funds $200 per million, then its equation is:

`100x + 200y lt= 6000`

And this simplifies to:

`x + 2y lt=60`

Hence, we have four constraints. These are:

EQ1: `x + y lt= 35`

EQ2: `x gt= 5`

EQ3: `ygt=15`

EQ4: `x + 2y lt= 60`

Next, let's determine the objective function. Since we have to maximize the annual interest for the two investments, apply the formula for simple interest.

`I=Prt`

where P is the principal amount (amount invested), r is the annual rate and t is the number of years.

So, the interest earned in each investment for one year is:

`I_x=x*0.05*1 = 0.05x`


`I_y=y*0.08*1=0.08y`

And, the annual interest earned for the two investments is:

`I_y+I_x=0.05x + 0.08y`

Hence, our objective function is:

`z=0.05x + 0.08y`

To determine the values of x and y that would maximize our z, graph the four inequality equations. Then, consider only the region that satisfy the four inequalities. 

Hence, is the region that satisfy the four inequalities is:

   

(Note: Green - boundary line of  EQ1.   Red - boundary line of EQ2.  

Orange - boundary line of EQ3    Blue - boundary line of EQ4.)

In our bounded region, the intersection of the boundary lines are (5,15) , (20,15), (10,25) and (5,27.5).

Plug-in these intersection point to the objective function

`z= 0.05x + 0.08y`

For (5,15), the value of z will be:

`z=0.05(5) + 0.08(15) = 1.45`

For (20,15), the value of z will be:

`z=0.05(20)+0.08(15)=2.2`

For (10,25), the value of z will be:

`z=0.05(10)+0.08(25)=2.5`

And for (5,27.5), the value of z will be:

`z=0.05(5) + 0.08(27.5)=2.45`

Hence, the values of x and y that will maximize z are x=10 and y=25

 

Therefore, to maximize the annual interest earned, the manager should invest $10,000,000 in bonds paying 5% annual interest and  $25,000,000 in mutual funds paying 8% annual interest.

 

 

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