This is your lucky day. You have won a $20,000 prize. You are setting aside $8,000 for taxes and partying expenses, but you have decided to invest the other $12,000. Upon hearing the news, two different friends have offered you an opportunity to become a partner in two different entrepreneurial ventures, one planned by each friend. In both cases, this investment would involve spending some of your time next summer as well as putting up cash. Becoming a full partner in the first friend’s venture would require an investment of $10,000 and 400 hours, and your estimated profit (ignoring the value of your time) would be $9,000. The corresponding figures for the second friend’s venture are $8,000 and 500 hours, with an estimated profit to you of $9,000. However, both friends are flexible and would allow you to come in at any fraction of a full partnership you would like. If you choose a fraction of a full partnership, all the above figures given for a full partnership (money investment, time investment, and your profit) would be multiplied by this fraction. Because you were looking for an interesting summer job anyway (maximum 600 hours), you have decided to participate in one or both friends’ ventures in whichever combination that would maximize your total estimated profit. You need to solve the problem of finding the best combination.

a) Use the graphical solution method to solve this problem. Clearly display the feasible region of the problem and its optimal solution.

b) What profit would the first friend have to offer you in order to be optimal to invest your money and time to become his full partner?