In Exercises \(I-8,\) write down (without solving) the dual LP problem. $$ \begin{array}{ll} \text { Maximize } & p=2 x+y \\ \text { subject to } & x+2 y \leq 6 \\ & -x+y \leq 2 \\ & x \geq 0, y \geq 0 . \end{array} $$

Gerber Mixed Cereal for Baby contains, in each serving, 60 calories and 11 grams of carbohydrates. Gerber Mango Tropical Fruit Dessert contains, in each serving, 80 calories and 21 grams of carbohydrates. \({ }^{3}\) You want to provide your child with at least 140 calories and at least 32 grams of carbohydrates. Draw the feasible region that shows the number of servings of cereal and number of servings of dessert that you can give your child. Find the corner points of the region.

Resource Allocation Fancy Pineapple produces pineapple juice and canned pineapple rings. This year the company anticipates a demand of at least 10,000 pints of pineapple juice and 1,000 cans of pineapple rings. Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple. The company anticipates using at least 20,000 pineapples for these products. Each pint of pineapple juice costs the company 20 e to produce, and each can of pineapple rings costs 50 \& to produce. How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs?

In Exercises 9-22, solve the given standard minimization problem using duality. (You may already have seen some of these in earlier sections, but now you will be solving them using \(a\) different method.) $$ \begin{array}{ll} \text { Minimize } & c=3 s+2 t \\ \text { subject to } & s+2 t \geq 20 \\ & 2 s+t \geq 10 \\ & s \geq 0, t \geq 0 . \end{array} $$

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

Based on the following data on three stocks: \(^{26}\) $$\begin{array}{|r|c|c|c|} \hline & \begin{array}{c}\text { Price } \\\\(\$)\end{array} & \begin{array}{c} \text { Dividend } \\\\\text { Yield }(\%)\end{array} & \begin{array}{c}\text { 52-Week } \\\\\text { Price Change }(\$)\end{array} \\\\\hline \begin{array}{r}\text { DUK } \\ (\text { Duke Energy Corp })\end{array} & 80 & 4 & 4 \\ \hline \text { DTV (DIRECTV) } & 100 & 0 & 10 \\\\\hline \begin{array}{r} \text { OCR } \\\\\text { (Omnicare, Inc.) }\end{array} & 90 & 1 & 30 \\\\\hline \end{array}$$ Repeat Exercise 37 under the assumption that the 52-week change in DTV stock is \(\$ 30\) but its price is unchanged.