Chapter 4

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 e per ounce and grain costs \(1 \phi\) per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost? HINT [See Example 4.]

$$ P=\left[\begin{array}{rrr} -1 & 1 & 2 \\ 2 & -1 & -2 \end{array}\right] $$

Agriculture \(^{29}\) Your small farm encompasses 100 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: \(\$ 5\) for tomatoes, \(\$ 4\) for lettuce, and \(\$ 2\) for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of \(\$ 2,000\) for each acre of tomatoes, \(\$ 1,500\) for each acre of lettuce and \(\$ 500\) for each acre of carrots. You would like to spend at least \(\$ 400\) on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 500 hours per week. How many acres of each crop should you plant to maximize total profits? In this event, will you be using all 100 acres of your farm? HINT [See Example 3.]

Resource Allocation Arctic Juice Company makes three juice blends: PineOrange, using 2 portions of pineapple juice and 2 portions of orange juice per gallon; PineKiwi, using 3 portions of pineapple juice and 1 portion of kiwi juice per gallon; and OrangeKiwi, using 3 portions of orange juice and 1 portion of kiwi juice per gallon. Each day the company has 800 portions of pineapple juice, 650 portions of orange juice, and 350 portions of kiwi juice available. Its profit on PineOrange is \(\$ 1\) per gallon, its profit on PineKiwi is \(\$ 2\) per gallon, and its profit on OrangeKiwi is \(\$ 1\) per gallon. How many gallons of each blend should it make each day to maximize profit? What is the largest possible profit the company can make?

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} -3 x+2 y & \leq 5 \\ 3 x-2 y & \leq 6 \\ x & \leq 2 y \\ x \geq 0, y \geq & 0 \end{aligned} $$

Enormous State University's Business School is buying computers. The school has two models from which to choose, the Pomegranate and the iZac. Each Pomegranate comes with \(400 \mathrm{MB}\) of memory and \(80 \mathrm{~GB}\) of disk space; each iZac has \(300 \mathrm{MB}\) of memory and \(100 \mathrm{~GB}\) of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least \(48,000 \mathrm{MB}\) of memory and at least \(12,800 \mathrm{~GB}\) of disk space. If the Pomegranate and the iZac cost \(\$ 2,000\) each, how many of each should the school buy to keep the cost as low as possible? HINT [See Example 4.]

$$ P=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 1 & 2 & 0 \end{array}\right] $$

Agriculture \(^{30}\) Your farm encompasses 900 acres, and you are planning to grow soybeans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: \(\$ 5\) for soybeans, \(\$ 2\) for corn, and \(\$ 1\) for wheat. You estimate that each acre of soybeans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of \(\$ 3,000\) for each acre of soybeans, \(\$ 2,000\) for each acre of corn, and \(\$ 1,000\) for each acre of wheat. You can afford to spend no more than \(\$ 3,000\) on fertilizer, but your labor union contract stipulates at least 2,000 hours per week of labor. How many acres of each crop should you plant to maximize total profits? In this event, will you be using more than 2,000 hours of labor?

Purchasing Trans Global Tractor Trailers has decided to spend up to \(\$ 1,500,000\) on a fleet of new trucks, and it is considering three models: the Gigahaul, which has a capacity of 6,000 cubic feet and is priced at \(\$ 60,000 ;\) the Megahaul, with a capacity of 5,000 cubic feet and priced at \(\$ 50,000\); and the Picohaul, with a capacity of 2,000 cubic feet, priced at \(\$ 40,000 .\) The anticipated annual revenues are \(\$ 500,000\) for each new. truck purchased (regardless of size). Trans Gilobal would like a total capacity of up to 130,000 cubic feet, and feels that it cannot provide drivers and maintenance for more than 30 trucks. How many of each should it purchase to maximize annual revenue? What is the largest possible revenue it can make?

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} -3 x+2 y & \leq 5 \\ 3 x-2 y & \geq 6 \\ y & \leq x / 2 \\ x \geq 0, y \geq & 0 \end{aligned} $$

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. \({ }^{11}\) If the cereal costs \(30 \phi\) per serving and the dessert costs 50 per serving, and you want to provide your child with at least 140 calories and at least 32 grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted.)

$$ P=\left[\begin{array}{rrr} -1 & 1 & 2 \\ 2 & -1 & -2 \\ 1 & 2 & 0 \end{array}\right] $$

Politics The political pollster Canter is preparing for a national election. It would like to poll at least 1,500 Democrats and 1,500 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost \(\$ 40\) each to produce and mail, mailings to the Midwest cost \(\$ 60\) each, and mailings to the West Coast cost \(\$ 50\) each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost? What will it cost?

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{array}{r} 2 x-y \geq 0 \\ x-3 y \leq 0 \\ x \geq 0, y \geq 0 \end{array} $$

Each serving of Gerber Mixed Cereal for Baby contains 60 calories, 10 grams of carbohydrates, and no vitamin \(\mathrm{C}\). Each serving of Gerber Apple Banana Juice contains 60 calories, 15 grams of carbohydrates, and 120 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. The cereal costs \(10 \mathrm{c}\) per serving and the juice costs \(30 \mathcal{c}\) per serving. If you want to provide your child with at least 120 calories, at least 25 grams of carbohydrates, and at least 60 percent of the U.S. RDA of vitamin C for infants, how can you do so at the least cost? (Fractions of servings are permitted.)

$$ P=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 1 & 2 & 0 \\ 0 & 1 & 1 \end{array}\right] $$

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 least possible cost?

Resource Allocation You manage an ice cream factory that makes three flavors: Creamy Vanilla, Continental Mocha, and Succulent Strawberry. Into each batch of Creamy Vanilla go 2 eggs, 1 cup of milk, and 2 cups of cream. Into each batch of Continental Mocha go 1 egg, 1 cup of milk, and 2 cups of cream. Into each batch of Succulent Strawberry go 1 egg, 2 cups of milk, and 2 cups of cream. You have in stock 200 eggs, 120 cups of milk, and 200 cups of cream. You make a profit of \(\$ 3\) on each batch of Creamy Vanilla, \(\$ 2\) on each batch of Continental Mocha and \(\$ 4\) on each batch of Succulent Strawberry. a. How many batches of each flavor should you make to maximize your profit? b. In your answer to part (a), have you used all the ingredients? c. Due to the poor strawberry harvest this year, you cannot make more than 10 batches of Succulent Strawberry. Does this affect your maximum profit?

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} -x+y & \geq 0 \\ 4 x-3 y & \geq 0 \\ x \geq 0, y & \geq 0 \end{aligned} $$

You are thinking of making your home more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs \(\$ 4\) and saves you an average of \(\$ 2\) per year in energy costs, and each square foot of wall insulation costs \(\$ 1\) and saves you an average of \(\$ 0.20\) per year in energy costs. \(^{12}\) Your home has 60 light fittings and 1,100 sq. ft. of uninsulated exterior wall. You can spend no more than \(\$ 1,200\) and would like to save as much per year in energy costs as possible. How many compact fluorescent light bulbs and how many square feet of insulation should you purchase? How much will you save in energy costs per year?

$$ P=\left[\begin{array}{rrrr} -1 & 1 & 2 & -1 \\ 2 & -1 & -2 & -3 \\ 1 & 2 & 0 & 1 \\ 0 & 2 & 3 & 3 \end{array}\right] $$

Resource Allocation Succulent Citrus produces orange juice and orange concentrate. This year the company anticipates a demand of at least 10,000 quarts of orange juice and 1,000 quarts of orange concentrate. Each quart of orange juice requires 10 oranges, and each quart of concentrate requires 50 oranges. The company also anticipates using at least 200,000 oranges for these products. Each quart of orange juice costs the company \(50 \varnothing\) to produce, and each quart of concentrate costs \(\$ 2.00\) to produce. How many quarts of each product should Succulent Citrus produce to meet the demand and minimize total costs?

Agriculture Your small farm encompasses 100 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: \(\$ 5\) for tomatoes, \(\$ 4\) for lettuce, and \(\$ 2\) for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of \(\$ 2,000\) for each acre of tomatoes, \(\$ 1,500\) for each acre of lettuce, and \(\$ 500\) for each acre of carrots. You can afford to spend no more than \(\$ 400\) on fertilizer, and your farm laborers can supply up to 500 hours per week. How many acres of each crop should you plant to maximize total profits? In this event, will you be using all 100 acres of your farm?

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ 2.1 x-4.3 y \geq 9.7 $$

(Compare with the preceding exercise.) You are thinking of making your mansion more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs \(\$ 4\) and saves you an average of \(\$ 2\) per year in energy costs, and each square foot of wall insulation costs \(\$ 1\) and saves you an average of \(\$ 0.20\) per year in energy costs. \(^{13}\) Your mansion has 200 light fittings and 3,000 sq. ft. of uninsulated exterior wall. To impress your friends, you would like to spend as much as possible, but save no more than \(\$ 800\) per year in energy costs (you are proud of your large utility bills). How many compact fluorescent light bulbs and how many square feet of insulation should you purchase? How much will you save in energy costs per year?

Resource Allocation Meow makes cat food out of fish and cornmeal. Fish has 8 grams of protein and 4 grams of fat per ounce, and cornmeal has 4 grams of protein and 8 grams of fat. A jumbo can of cat food must contain at least 48 grams of protein and 48 grams of fat. If fish and cornmeal both cost 5elounce, how many ounces of each should Meow use in each can of cat food to minimize costs? What are the shadow costs of protein and of fat? HINT [See Example 2.]

Agriculture Your farm encompasses 500 acres, and you are planning to grow soybeans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: \(\$ 5\) for soybeans, \(\$ 2\) for corn, and \(\$ 1\) for wheat. You estimate that each acre of soybeans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of \(\$ 3,000\) for each acre of soybeans, \(\$ 2,000\) for each acre of corn, and \(\$ 1,000\) for each acre of wheat. You can afford to spend no more than \(\$ 3,000\) on fertilizer, and your farm laborers can supply 3,000 hours per week. How many acres of each crop should you plant to maximize total profits? In this event, will you be using all the available labor?

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ -4.3 x+4.6 y \geq 7.1 $$

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ \begin{aligned} -0.2 x+0.7 y & \geq 3.3 \\ 1.1 x+3.4 y & \geq 0 \end{aligned} $$

Music CD Sales Your music store's main competitor, Nuttal Hip Hop Classic Store, also wishes to stock at most 20,000 CDs, with at least half as many rap CDs as rock CDs and at least 2,000 classical CDs. It anticipates an average sale price of \(\$ 15 /\) rock CD, \$10/rap CD and \$10/classical CD. How many of each type of CD should it stock to get the maximum retail value, and what is the maximum retail value?

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ \begin{aligned} &0.2 x+0.3 y \geq 7.2 \\ &2.5 x-6.7 y \leq 0 \end{aligned} $$

Your salami manufacturing plant can order up to 1,000 pounds of pork and 2,400 pounds of beef per day for use in manufacturing its two specialties: "Count Dracula Salami" and "Frankenstein Sausage." Production of the Count Dracula variety requires 1 pound of pork and 3 pounds of beef for each salami, while the Frankenstein variety requires 2 pounds of pork and 2 pounds of beef for every sausage. In view of your heavy investment in advertising Count Dracula Salami, you have decided that at least onethird of the total production should be Count Dracula. On the other hand, due to the health-conscious consumer climate. your Frankenstein Sausage (sold as having less beef) is earning your company a profit of \(\$ 3\) per sausage, while sales of the Count Dracula variety are down and it is earning your company only \(\$ 1\) per salami. Given these restrictions, how many of each kind of sausage should you produce to maximize profits, and what is the maximum possible profit? HINT [See Example 3.]

Nutrition Ruff makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat/ounce. and grain has 2 grams of protein and 2 grams of fat/ounce. \(\mathrm{A}\) bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 ec/ounce and grain costs 1 d/ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost? What are the shadow costs of protein and fat?

I Nutrition Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of vitamin \(\mathrm{C}\) for infants. \(^{32}\) The cereal costs \(10 \mathrm{~d} / \mathrm{serving}\), the dessert costs \(53 \mathrm{~d} /\) serving, and the juice costs \(27 \mathrm{c} /\) serving. If you want to provide your child with at least 120 calories and at least 120 percent of the RDA of vitamin \(\mathrm{C}\), how can you do so at the least cost?

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ \begin{aligned} &4.1 x-4.3 y \leq 4.4 \\ &7.5 x-4.4 y \leq 5.7 \\ &4.3 x+8.5 y \leq 10 \end{aligned} $$

The Megabuck Hospital Corp. is to build a state-subsidized nursing home catering to homeless patients as well as high-income patients. State regulations require that every subsidized nursing home must house a minimum of 1,000 homeless patients and no more than 750 high-income patients in order to qualify for state subsidies. The overall capacity of the hospital is to be 2,100 patients. The board of directors, under pressure from a neighborhood group, insists that the number of homeless patients should not exceed twice the number of high-income patients. Due to the state subsidy, the hospital will make an average profit of \(\$ 10,000\) per month for every homeless patient it houses, whereas the profit per high-income patient is estimated at \(\$ 8,000\) per month. How many of each type of patient should it house in order to maximize profit? HINT [See Example 3.]

We suggest you use technology. Graph the regions corresponding to the inequalities, and find the coordinates of all corner points (if any) to two decimal places: $$ \begin{aligned} &2.3 x-2.4 y \leq 2.5 \\ &4.0 x-5.1 y \leq 4.4 \\ &6.1 x+6.7 y \leq 9.6 \end{aligned} $$

In February 2008 , each episode of "American Idol" was typically seen by \(28.5\) million viewers, while each episode of "Back to You" was seen by \(12.3\) million viewers. \(^{15}\) Your marketing services firm has been hired to promote Bald No More's hair replacement process by buying at least 30 commercial spots during episodes of "American Idol" and "Back to You." The cable company running "American Idol" has quoted a price of \(\$ 3,000\) per spot, while the cable company showing "Back to You" has quoted a price of \(\$ 1,000\) per spot. Bald No More's advertising budget for TV commercials is \(\$ 120,000\), and it would like no more than \(50 \%\) of the total number of spots to appear on "Back to You." How many spots should you purchase on each show to reach the most viewers?

I Purchasing Cheapskate Electronics Store needs to update its inventory of stereos, TVs, and DVD players. There are three suppliers it can buy from: Nadir offers a bundle consisting of 5 stereos, 10 TVs, and 15 DVD players for \(\$ 3,000\). Blunt offers a bundle consisting of 10 stereos, \(10 \mathrm{TVs}\), and 10 DVD players for \(\$ 4,000\). Sonny offers a bundle consisting of 15 stereos, \(10 \mathrm{TVs}\), and 10 DVD players for \(\$ 5,000\). Cheapskate Electronics needs at least 150 stereos, \(200 \mathrm{TVs}\), and 150 DVD players. How can it update its inventory at the least possible cost? What is the least possible cost?

Y o u ~ m a n a g e ~ a n ~ i c e ~ c r e a m ~ f a c t o r y ~ t h a t ~ makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. Draw the feasible region showing the number of quarts of vanilla and number of quarts of mocha that can be produced. Find the corner points of the region. HINT [See Example 4.1

In February 2002, each episode of "Boston Public" was typically seen in \(7.0\) million homes, while each episode of "NYPD Blue" was seen in \(7.8\) million homes. \({ }^{16}\) Your marketing services firm has been hired to promote Gauss Jordan Sneakers by buying at least 30 commercial spots during episodes of "Boston Public" and "NYPD Blue." The cable company running "Boston Public" has quoted a price of \(\$ 2,000\) per spot, while the cable company showing "NYPD Blue" has quoted a price of \(\$ 3,000\) per spot. Gauss Jordan Sneakers' advertising budget for TV commercials is \(\$ 70,000\), and it would like at least \(75 \%\) of the total number of spots to appear on "Boston Public." How many spots should you purchase on each show to reach the most homes?

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no vitamin \(C\). Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\) for infants. \(^{42}\) The cereal costs \(10 \mathrm{~d} /\) serving, the dessert costs \(53 \mathrm{~d} /\) serving, and the juice costs 27 d/serving. If you want to provide your child with at least 120 calories and at least 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\), how can you do so at the least cost? What are your shadow costs for calories and vitamin \(\mathrm{C}\) ?

f Purchasing Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for \(\$ 500,000\). Admiral Motors is offering 5 small, 10 medium, and 5 large cars for \(\$ 400,000\). Chrysalis is offering 10 small, 5 medium, and 5 large cars for \(\$ 300,000\). Federal would like to buy at least 550 small cars, at least 500 medium cars, and at least 550 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible? What will be the total cost?

Podunk Institute of Technology's Math Department offers two courses: Finite Math and Applied Calculus. Each section of Finite Math has 60 students, and each section of Applied Calculus has \(50 .\) The department is allowed to offer a total of up to 110 sections. Furthermore, no more than 6,000 students want to take a math course. (No student will take more than one math course.) Draw the feasible region that shows the number of sections of each class that can be offered. Find the corner points of the region. HINT [See Example 4.]

The political pollster Canter is preparing for a national election. It would like to poll at least 1,500 Democrats and 1,500 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost \(\$ 40\) each to produce and mail, mailings to the Midwest cost \(\$ 60\) each, and mailings to the West Coast cost \(\$ 50\) each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost? What will it cost? What are the shadow costs of a Democratic response and a Republican response?

You are planning to invest up to \(\$ 10,000\) in MON, SNE, and IMCL shares. You desire to maximize your share of the companies' earnings but, for tax reasons, want to earn no more than \(\$ 200\) in dividends. Your broker suggests that because IMCL stock pays no dividends, you should invest only in IMCL. Is she right?

Ruff Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. Draw the feasible region that shows the number of ounces of chicken and number of ounces of grain Ruff can mix into each bag of dog food. Find the corner points of the region.

$$ \begin{array}{|r|c|c|c|c|c|} \hline & \begin{array}{c} \text { Creatine } \\ (g) \end{array} & \begin{array}{c} \text { Carbohydrates } \\ (g) \end{array} & \begin{array}{c} \text { Taurine } \\ (g) \end{array} & \begin{array}{c} \text { Alpha Lipoic } \\ \text { Acid }(\mathrm{mg}) \end{array} & \begin{array}{c} \text { Cost } \\ (\$) \end{array} \\ \hline \begin{array}{r} \text { Cell-Tech }^{\infty} \\ \text { (MuscleTech) } \end{array} & 10 & 75 & 2 & 200 & 2.20 \\ \hline \begin{array}{r} \text { RiboForce } \\ \mathbf{H P}^{*} \text { (EAS) } \end{array} & 5 & 15 & 1 & 0 & 1.60 \\ \hline \begin{array}{r} \text { Creatine } \\ \text { Transport }^{*} \\ \text { (Kaizen) } \end{array} & 5 & 35 & 1 & 100 & 0.60 \\ \hline \begin{array}{r} \text { Pre-Load } \\ \text { Creatine } \\ \text { (Optimum) } \end{array} & 6 & 35 & 1 & 25 & 0.50 \\ \hline \end{array} $$ (Compare Exercise 30 in Section 4.2.) You are thinking of combining RiboForce HP, Creatine Transport, and Pre-Load Creatine to obtain a 10 -day supply that provides at least 80 grams of creatine and at least 10 grams of taurine, but no more than 600 grams of carbohydrates and 2,000 milligrams of alpha lipoic acid. How many servings of each supplement should you combine to meet your specifications for the least cost?

Enormous State University's Business School is buying computers. The school has two models to choose from: the Pomegranate and the iZac. Each Pomegranate comes with \(400 \mathrm{MB}\) of memory and \(80 \mathrm{~GB}\) of disk space, and each iZac has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least \(48,000 \mathrm{MB}\) of memory and at least \(12,800 \mathrm{~GB}\) of disk space. Draw the feasible region that shows the number of each kind of computer it can buy. Find the corner points of the region.

Investments Your portfolio manager has suggested two high-yielding stocks: Consolidated Edison (ED) and General Electric (GE). ED shares cost \(\$ 40\), yield \(6 \%\) in dividends, and have a risk index of \(2.0\) per share. GE shares cost \(\$ 16\), yield \(7.5 \%\) in dividends, and have a risk index of \(3.0\) per share. \({ }^{18}\) You have up to \(\$ 10,000\) to invest, and would like to earn at least \$600 in dividends. How many shares (to the nearest tenth of a unit) of each stock should you purchase to meet your requirements and minimize the total risk index for your portfolio? What is the minimum total risk index?