George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: $(1)$ a bond

fund and $(2)$ a stock fund. The projected returns over the life of the investments are $7\%$ for the bond fund and $13\%$ for the stock fund. Whatever portion of the inheritance he finally decides to commit

to the trust fund, he wants to invest at least $30\%$ of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least $8\%.$

$($a$)$ Formulate a linear programming model that can be used to determine the percentage $($as a decimal$)$ that should be allocated to each of the possible investment alternatives. $($Assume $B$ is the

percentage $($as a decimal$)$ of funds invested in the bond fund and $S$ is the percentage $($as a decimal$)$ of funds invested in the stock fund.$)$

Max $($as a decimal$)$

s$.$t$.$

bond minimum fund

minimum return

percentage requirement

$B,S0$

$($b$)$ Solve the problem using the graphical solution procedure. What is the value of the objective function at the optimal solution?

$at(B,S)=(,)$