Q327. Our agency provides three types of client service: A, B, and C. And we have 3 kinds of staff: X, Y, and Z.

Each type A service requires 3 hours of an X staff member's time and 1 hour of a Y. Type B requires 2 X, 1 Y, and 3 Z hours. And type C requires 1 X, 3 Y, and 2 Z.

Currently we have 2 X, 1 Y and 1 Z on staff. We pay X's \$25 per hour, Ys get \$30 and Zs get \$40. Assume everyone works a 35 hour week. At 35 hours per week our labor costs are 4200.

Revenue from type A service is \$100, B is \$200, and C is \$300.

Regulations require that we serve at least 5 of each client type each week and that we serve at total of at least 21 clients each week.

What client mix will allow us to maximize revenue?

Excel Worksheet here.

Q231. A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 20 m3 and a non-refrigerated capacity of 40 m3 while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of 3,000 m3 of refrigerated stock and 4 000 m3 of non-refrigerated stock. The cost per kilometer of a Type A is \$30, and \$40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

Alternatively

A school district has two types of lower division schools, type A and type B. Type A school buildings have capacity for 200 little kids and 400 big kids. Type B buildings have capacity for 300 little kids and 300 big kids. Next year the district expects enrollments of 3000 little kids and 4000 big kids. Type A buildings cost 30,000 per year to maintain while type B buildings cost 40,000. What mix of school buildings will allow the district to handle the expected enrollment at the lowest maintenance cost? (From VITutor)

Q230. A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for \$30. Offer B is a package of three shirts and a pair of pants, which will sell for \$50. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to maximize the money generated from the promotion? (From VITutor)

Q229. Bob builds tool sheds. He uses 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes \$390 profit on a small shed and \$520 on a large shed, how many of each type of building should Bob build to maximize his profit? (From solution here)

Q228. A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is \$800 and \$600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost. (From VITutor)

Q227. A gold processor has two sources of gold ore, source A and source B. In order to kep his plant running, at least three tons of ore must be processed each day. Ore from source A costs \$20 per ton to process, and ore from source B costs \$10 per ton to process. Costs must be kept to less than \$80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints? (From Steve Wilson)

Q226. "You have \$12,000 to invest, and three different funds from which to choose. The municipal bond fund has a 7% return, the local bank's CDs have an 8% return, and the high-risk account has an expected (hoped-for) 12% return. To minimize risk, you decide not to invest any more than \$2,000 in the high-risk account. For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the year-end yields are as expected, what are the optimal investment amounts?" (From PurpleMath.com)

Q225. Your are the supervisor at a new after-school program. The program will serve 100 boys and 100 girls. Activities will include chess, games, and crafts. Materials, supervision, and the like have been priced out at \$2/person for chess, \$10/person for games, and \$5 for crafts. Space needs are such that we can get 8 chess players at a table, 4 games players, or 2 crafters. The center has 50 tables. Solid research has shown that activity preferences among this population of children is somewhat gender specific. Boys and girls like chess the same but games are 70% girls and 30% boys while crafts tend to be 30% girls and 70% boys. What is the most economical division of activities subject to these constraints?

Q224. You are working for an agricultural cooperative which is helping local farmers figure out how to optimize the mixture of crops they plant. A typical farmer has 10 acres to plant in wheat and rye. She has to plant at least 7 acres. However, she has only the equivalent of \$1200 to spend and each acre of wheat costs \$200 to plant and each acre of rye costs \$100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the expected profit is \$500 per acre of wheat and \$300 per acre of rye how many acres of each should be planted to maximize profits? (From Steve Wilson)

Q223. A non-profit supplier of after-school materials has orders for 600 copies from San Francisco and 400 copies from Sacramento. The organization has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs \$5 to ship a text from Novato to San Francisco, but it costs \$10 to ship it to Sacramento. It costs \$15 to ship from Lodi to San Francisco, but it costs \$4 to ship it from Lodi to Sacramento. How many copies should the organization ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost? [http://www.sonoma.edu/users/w/wilsonst/Courses/Math_131/lp/default.html]

Q222. Write out the sample problems on pp 190-2 with explanatory solutions.

Q221. For each of the problems described below, say whether it is best thought of as an analog to diet, transport, activity, or assignment as outlined above.

1. S&Z problem #1 Incinerators DIET TRANSPORT ACTIVITY ASSIGNMENT
2. S&Z problem #2 Police Shifts DIET TRANSPORT ACTIVITY ASSIGNMENT
3. S&Z problem #3 Hospitals and disasters DIET TRANSPORT ACTIVITY ASSIGNMENT
4. S&Z problem #4 Electricity generation and pollution DIET TRANSPORT ACTIVITY ASSIGNMENT
5. S&Z text example – transit maintenance DIET TRANSPORT ACTIVITY ASSIGNMENT

Q220. What is the objective function in each of the following situations?

1. What is the largest volume box I can make by folding a piece of cardboard that is A inches by B inches?
2. Pancakes cost \$1 each, eggs are 1.50, and blintzes are 2. Pancakes have 200 calories, eggs 125 and blintzes 450. What combination gives me the most calories for 5 dollars?
3. What's the cheapest 1000 calorie daily diet?
4. I have information on the level of AOD demand reduction we can expect from public awareness campaigns, DARE visits to public schools, increased treatment slots, and increases in after care. I know the cost of each type program and I have a limited budget. What mix of programs should I institute to have the biggest effect on demand?

Q219. Translate the following into inequalities.

1. I can only I spend as much cash as in my wallet on dinner, dessert, drinks, and a tip and I really want to have dinner and drinks though I might pass on dessert.
2. You are managing a youth shelter. Kids present with an array of personal challenges, each of which require different levels of attention from your staff. Clients with issue A require 4 hours of attention per week. Issue B, about 2 hours, C requires 16, and D 7. Your budget allows you to staff 75 hours per week.
3. Breakfast is some eggs, some pancakes, some bacon. You have to have at least twice as many pancakes as eggs. You can't have fewer than 2 strips of bacon.

#### From PurpleMath.com

"In order to ensure optimal health (and thus accurate test results), a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protien. But the rabbits should be fed no more than five ounces of food a day.

Rather than order rabbit food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs \$0.20 per ounce. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of \$0.30 per ounce.

What is the optimal blend?"

#### From PurpleMath.com

"You have \$12,000 to invest, and three different funds from which to choose. The municipal bond fund has a 7% return, the local bank's CDs have an 8% return, and the high-risk account has an expected (hoped-for) 12% return. To minimize risk, you decide not to invest any more than \$2,000 in the high-risk account. For tax reasons, you need to invest at least three times as much in the municipal bonds as in the bank CDs. Assuming the year-end yields are as expected, what are the optimal investment amounts?"

#### From Steve Wilson

A publisher has orders for 600 copies of a certain text from San Francisco and 400 copies from Sacramento. The company has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs \$5 to ship a text from Novato to San Francisco, but it costs \$10 to ship it to Sacramento. It costs \$15 to ship a text from Lodi to San Francisco, but it costs \$4 to ship it from Lodi to Sacramento. How many copies should the company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost?

#### From Steve Wilson

A gold processor has two sources of gold ore, source A and source B. In order to kep his plant running, at least three tons of ore must be processed each day. Ore from source A costs \$20 per ton to process, and ore from source B costs \$10 per ton to process. Costs must be kept to less than \$80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints?

#### From Steve Wilson

A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only \$1200 to spend and each acre of wheat costs \$200 to plant and each acre of rye costs \$100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is \$500 per acre of wheat and \$300 per acre of rye how many acres of each should be planted to maximize profits?

#### From Mona Baarson

A dietitian wants to design a breakfast menu for certain hospital patients. The menu is to include two items A and B. Suppose that each ounce of A provides 2 units of vitamin C and 2 units of iron and each ounce of B provides 1 unit of vitamin C and 2 units of iron. Suppose the cost of A is 4¢/ounce and the cost of B is 3¢/ounce. If the breakfast menu must provide at least 8 units of vitamin C and 10 units of iron, how many ounces of each item should be provided in order to meet the iron and vitamin C requirements for the least cost? What will this breakfast cost"

#### From Algebra Lab

Bob builds tool sheds. He uses 10 sheets of dry wall and 15 studs for a small shed and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If Bob makes \$390 profit on a small shed and \$520 on a large shed, how many of each type of building should Bob build to maximize his profit?

#### From VITutor

A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 20 m3 and a non-refrigerated capacity of 40 m3 while Type B has the same overall volume with equal sections for refrigerated and non-refrigerated stock. A grocer needs to hire trucks for the transport of 3,000 m3 of refrigerated stock and 4 000 m3 of non-refrigerated stock. The cost per kilometer of a Type A is \$30, and \$40 for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

Alternatively

A school district has two types of lower division schools, type A and type B. Type A school buildings have capacity for 200 little kids and 400 big kids. Type B buildings have capacity for 300 little kids and 300 big kids. Next year the district expects enrollments of 3000 little kids and 4000 big kids. Type A buildings cost 30,000 per year to maintain while type B buildings cost 40,000. What mix of school buildings will allow the district to handle the expected enrollment at the lowest maintenance cost?

#### From VITutor

A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is \$800 and \$600 for the small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

#### From VITutor

A store wants to liquidate 200 of its shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for \$30. Offer B is a package of three shirts and a pair of pants, which will sell for \$50. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to sell to maximize the money generated from the promotion?