An investor is considering two types of investment. One option is a high$-$risk venture into oil leases with a potential return of $15\%,$ while the second option is a relatively safe stock investment with an $8\%$ return rate. He has $$50,000$ to invest. Because of the risk, he would like to invest no more than $40\%$ of his total investment in the oil leases, and at least $50\%$ of the total investment in the stock. How much should he invest in each of the options so as to maximize his total return? Formulate an LP model for the problem. $9$ marks

Investment Options

Oil Leases Total amount for investment $1$

Return rate $15\%$

Safe Stock

$1$ $$50,000$

$8\%$ Maximize

Upper limit on oil leases

No more than $40\%$ of the total amount invested

Lower limit on safe stock

At least $50\%$ of the total amount invested