Question

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 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?

The amount that should be invested in Treasury bonds is ​$___ million and the amount that should be invested in mutual funds is ​$__ million.

The maximum annual interest is ​$___

Answer 1

Let x be the amount invested in Treasury bonds and y be the amount invested in mutual funds. Then we have the following constraints:

x + y ≤ 40 (total investment cannot exceed $40 million) x ≥ 5 (at least $5 million must be invested in Treasury bonds) y ≥ 15 (at least $15 million must be invested in mutual funds) 100x + 200y ≤ 7000 (total fees cannot exceed $7000)

The objective is to maximize the annual interest, which is given by:

0.06x + 0.08y

We can solve this problem using linear programming. The feasible region is a polygon with vertices at (5, 15), (5, 35), (30, 10), and (40, 0). We evaluate the objective function at each vertex:

At (5, 15): 0.06(5) + 0.08(15) = 1.2At (5, 35): 0.06(5) + 0.08(35) = 2.6At (30, 10): 0.06(30) + 0.08(10) = 2.8At (40, 0): 0.06(40) + 0.08(0) = 2.4

The maximum value of the objective function is 2.8, which occurs at (30, 10). Therefore, the amount that should be invested in Treasury bonds is $30 million and the amount that should be invested in mutual funds is $10 million.

The annual interest earned is 0.06(30) + 0.08(10) = $2.8 million.

Explanation: