Chapter 4.1, Problem 47E

ASSET ALLOCATION Sharon has a total of $\$200,000$ to invest in three types of mutual funds: growth, balanced, and income funds.Growth funds have a rate of return of $12\%/\text{year}$, balanced funds have rate of return of $10\%/\text{year}$, and income funds have a return of $6\%/\text{year}$.The growth, balanced, and income mutual funds are assigned risk factors of $0.1$, $0.06$, and $0.02$, respectively. Sharon has decided that at least $50\%$ of her total portfolio is to be in income funds and at least $25\%$ in balanced funds.She has also decided that the average risk factor for her investment should not exceed $0.05$.How much should Sharon invest in each type of fund to realize a maximum return on her investment?What is the maximum return?

Hint: The constraint for the average risk factor for the investment is given by $0.1x+0.06y+0.02z\le 0.05\left(x+y+z\right)$.