# Linear optimization

• Oct 1st 2012, 07:47 AM
Niaboc
Linear optimization
A pension fund manager decides to invest a total of at most \$39 million in U.S. treasury bonds paying 4%
annual interest and in mutual funds paying 8% annual interest. He plans to invest at least \$5 million in
bonds and at least \$10 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
\$5000 on fees. How much should be invested in each to maximize annual interest? What is the maximum
annual interest?

How would i set up the max and min and then eventually graph it? and thus solve the problem
Thank you
• Oct 1st 2012, 10:14 AM
Soroban
Re: Trignomy word problem
Hello, Niaboc!

I'll set it up . . .

Quote:

A pension fund manager decides to invest a total of at most \$39 million
in Treasury Bonds paying 4% annual interest and in Mutual Funds paying 8% annual interest.
He plans to invest at least \$5 million in Trasury Bonds and at least \$10 million in Mutual Funds.
Treasury 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 \$5000 on fees.

How much should be invested in each to maximize annual interest?
What is the maximum annual interest?

Let \$\displaystyle x\$ = amount invested in Treasury Bonds (in millions of dollars): \$\displaystyle x\ge\,0\$
Let \$\displaystyle y\$ = amount invested in Mutual Funds (in millions of dollars): .\$\displaystyle y \,\ge\,0\$

Total invested, \$39 million: .\$\displaystyle x + y \:\le\:39\$

Invest at least \$10 million in Mutual Funds: .\$\displaystyle x \,\ge\,10\$
Invest at least \$5 million in Treasure Bonds: .\$\displaystyle y \,\ge\,5\$

Fee for Treasury Bonds: \$\displaystyle 100x\$
Fee for Mutual Funds: \$\displaystyle 200y\$
Maximum fee, \$5000: .\$\displaystyle 100x + 200y \,\le\,5000 \$

Maximize interest: .\$\displaystyle I \:=\:0.04x + 0.08y\$