Problem $1$

Jim is planning to invest between $$12,000$ and $$15,000$ in two types of investments: investment $1$ yields $6\%,$ and investment $2$ yields $8\%.$ Because of tax laws, he should invest between $40\%$ and $60\%$ of his total investment in investment $1$ and no more than $$8,100$ in investment $2.$ How much should he invest, and how should that amount be allocated to the two investments?

a$)$ In this document, describe the components $($decision variables, objective function, and constraints$)$ in the context of this problem.

b$)$ In this document, write the mathematical formulation of the problem.

c$)$ In this document, solve the problem graphically and explain the optimal allocation of investment funds to the two investments.

d$)$ In an Excel workbook, solve the problem by using the Solver.