Solve the given LP problem. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. $\begin{array}{ll}\text { Minimize } & c=0.4 x+0.1 y \\ \text { subject to } & 30 x+20 y \geq 600 \\ & 0.1 x+0.4 y \geq 4 \\ & 0.2 x+0.3 y \geq 4.5 \\ & x \geq 0, y \geq 0\end{array}$

What is a "basic solution"? How might one find a basic solution of a given system of linear equations?

Investments: Financial Stocks (Compare Exercise 51 in Section 3.1.) During the first quarter of 2015 , Toronto Dominion Bank (TD) stock cost \(\$ 45\) per share and was expected to yield \(4 \%\) per year in dividends, while CNA Financial Corp. (CNA) stock cost \$40 per share and was expected to yield $2.5 \%\( per year in dividends. \){ }^{7}\( You have up to \)\$ 25,000$ to invest in these stocks and would like to earn at least \(\$ 760\) in dividends over the course of a year. (Assume the dividend to be unchanged for the year.) Draw the feasible region that shows how many shares in each company you can buy. Find the corner points of the region. (Round each coordinate to the nearest whole number.)

Purchasing Bingo's Copy Center needs to buy white paper and yellow paper. Bingo's can buy from three suppliers. Harvard Paper sells a package of 20 reams of white and 10 reams of yellow for \(\$ 60 ;\) Yale Paper sells a package of 10 reams of white and 10 reams of yellow for \(\$ 40,\) and Dartmouth Paper sells a package of 10 reams of white and 20 reams of yellow for \(\$ 50 .\) If Bingo's needs 350 reams of white and 400 reams of yellow, how many packages should it buy from each supplier to minimize the cost? What is the lowest possible cost? What are the shadow costs of white paper and yellow paper?

Game Theory: Military Strategy Colonel Blotto is a well-known game in military strategy. \({ }^{45}\) Here is a version of this game: Colonel Blotto has four regiments under his command, while his opponent, Captain Kije, has three. The armies are to try to occupy two locations, and each commander must decide how many regiments to send to each location. The army that sends more regiments to a location captures that location as well as the other army's regiments. If both armies send the same number of regiments to a location, then there is a draw. The payoffs are one point for each location captured and one point for each regiment captured. Find the optimum strategy for each commander and also the value of the game.

In Exercises \(I-8,\) write down (without solving) the dual LP problem. $$ \begin{array}{ll} \text { Maximize } & p=x+y+z+w \\ \text { subject to } & x+y+z \leq 3 \\ & y+z+w \leq 4 \\ & x+z+w \leq 5 \\ & x+y+w \leq 6 \\ & x \geq 0, y \geq 0, z \geq 0, w \geq 0 \end{array} $$