An investor is considering three types of investments: a high$-$risk venture into oil leases with a potential return of $15\%,$ a medium$-$risk investment in bonds with a $9\%$ return, and a relatively safe stock investment with a $5\%$ return. He has $$50,000$ to invest. Because of the risk, he will limit his investment in oil leases and bonds to $30\%$ and his investment in oil leases and stock to $50\%.$ How much should he invest in each to maximize his return, assuming investment returns are as expected?

a$.)$ Define the variables. Be specific with descriptive words.

$x=$

$y=$

$z=$

b$.)$ Clearly state the constraints $($all inequalities$)$ related to the feasible region.

c$.)$ State the objective function.

d$.)$ Set up the initial simplex matrix needed to solve the linear programming problem using the Simplex Method.

e$.)$ perform all pivots necessary using row operations to transform the matrix until the solution is feasible

f$.)$ how much should he invest in each to maximize his return, assuming investment returns are as expected?