Q: A restaurant charges $$2.10 for a breakfast taco and a small glass of milk. The cost for 2 tacos and 3 small glasses of milk is 5.15. $$What is the cost of a taco and the cost of a small glass of milk?

$$1.15 \text{ for a taco and } \0.95$$ \text{ for a small glass of milk}$

Q: The Frosty Ice-Cream Shop sells sundaes for $$2 and banana splits for 3. On a $$hot summer day, the shop sold 8 more sundaes than banana splits and made $100. How many of each dessert were sold?

$$36 \text{ sundaes and } 28 \text{ banana splits}$$

Q: The perimeter of a rectangular wooden deck is 90 feet. The deck's length, $$l$$, is 5 feet less than 4 times its width, $$w. $$What are the dimensions of the deck?

$$10 \text{ ft } \times 35 \text{ ft}$$

Q: 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 he makes $$390 profit on a small shed and 520 on a $$large shed, how many of each type of shed should Bob build to maximize his profit?

$$3 \text{ small and } 2 \text{ large}$$

Q: A farmer wants to plant corn and wheat. He wants to plant no more than 120 acres of both crops. At least 20 acres of corn will be planted. He also wants to plant at least 20 acres of wheat, but no more than 60 acres of wheat. The farmer makes a profit of $$357.54 per acre for planting corn and 319.32 $$per acre for planting wheat. What is the objective function for the farmer's total profit and what is the maximum profit?

$$P(x, y) = 357.54x + 319.32y; \text{ Max profit: } \39$$,929.20$

Q: A farmer wants to plant wheat and rye. He wants to plant at least 12 acres of wheat. He has only $$1200 to spend and each acre of wheat costs 210 to $$plant and each acre of rye costs $$100 to plant. The farmer also has no more than 40 acres of land. If wheat yields 500 $$per acre and rye yields $300 per acre, how many acres of each should be planted to maximize profit?

$$4 \text{ acres of wheat and } 4 \text{ acres of rye}$$

Q: Toy wagons are made to sell at a craft fair. It takes 4 hours to make a small wagon and 6 hours to make a large wagon. The owner of the craft booth will make a profit of $$12 for a small wagon and 30 $$for a large wagon and has no more than 60 hours available to make wagons. The owner wants to have at least 6 small wagons to sell. How many of each size should be made to maximize profit?

$$6 \text{ small wagons and } 6 \text{ large wagons}$$

Q: Suppose a manufacturer makes two types of benches: Home-Fitter and Bomer-Fitter. Each Home-Fitter takes 4 hours to trim and ten hours to sand the lathe, and 2 hours to finish. Each Bomer-Fitter needs 5 hours to trim, 8 to sand, and 6 to finish. The available workers can provide 60 hours for trimming, 100 hours for sanding, and 36 hours for finishing. How many of each type of bench should be produced to make a maximum profit?

$$6 \text{ Home-Fitter and } 3 \text{ Bomer-Fitter}$$

Question 1

If the sum of two numbers is 12 and their difference is 4, what are the two numbers?

Accepted Answer

$$8 	ext{ and } 4$$

Question 2

A restaurant charges $$2.10 for a breakfast taco and a small glass of milk. The cost for 2 tacos and 3 small glasses of milk is 5.15. $$What is the cost of a taco and the cost of a small glass of milk?

Accepted Answer

$$1.15 	ext{ for a taco and } \0.95$$ 	ext{ for a small glass of milk}$

Question 3

The Frosty Ice-Cream Shop sells sundaes for $$2 and banana splits for 3. On a $$hot summer day, the shop sold 8 more sundaes than banana splits and made $100. How many of each dessert were sold?

Accepted Answer

$$36 	ext{ sundaes and } 28 	ext{ banana splits}$$

Question 4

The perimeter of a rectangular wooden deck is 90 feet. The deck's length, $$l$$, is 5 feet less than 4 times its width, $$w. $$What are the dimensions of the deck?

Accepted Answer

$$10 	ext{ ft } 	imes 35 	ext{ ft}$$

Question 5

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 he makes $$390 profit on a small shed and 520 on a $$large shed, how many of each type of shed should Bob build to maximize his profit?

Accepted Answer

$$3 	ext{ small and } 2 	ext{ large}$$

Question 6

A toy manufacturer wants to minimize cost for producing two lines of toy airplanes. Because of the supply of materials, no more than 40 Flying Bat can be built each day, and no more than 60 Flying Falcons can be built each day. There are enough workers to build at least 70 toy airplanes each day. It costs $$12 to manufacture a Flying Bat and 8 to $$build a Flying Falcon. What is the minimum production cost per day?

Accepted Answer

$$600$$

Question 7

A tree nursery has red and maple trees available for planting. A city has $$2100 to spend on planting trees in urban areas. The land available for planting is 45,000 square feet. Spruce trees cost 30 to $$plant and require 600 square feet of space. Maple trees cost $40 to plant and require 900 square feet of space. What is the maximum number of spruce trees that can be planted if no maple trees are planted?

Accepted Answer

$$60$$

Question 8

A farmer wants to plant corn and wheat. He wants to plant no more than 120 acres of both crops. At least 20 acres of corn will be planted. He also wants to plant at least 20 acres of wheat, but no more than 60 acres of wheat. The farmer makes a profit of $$357.54 per acre for planting corn and 319.32 $$per acre for planting wheat. What is the objective function for the farmer's total profit and what is the maximum profit?

Accepted Answer

$$P(x, y) = 357.54x + 319.32y; 	ext{ Max profit: } \39$$,929.20$

Question 9

A farmer wants to plant wheat and rye. He wants to plant at least 12 acres of wheat. He has only $$1200 to spend and each acre of wheat costs 210 to $$plant and each acre of rye costs $$100 to plant. The farmer also has no more than 40 acres of land. If wheat yields 500 $$per acre and rye yields $300 per acre, how many acres of each should be planted to maximize profit?

Accepted Answer

$$4 	ext{ acres of wheat and } 4 	ext{ acres of rye}$$

Question 10

A gold processor uses two sources of gold ore, source A and source B. In order to keep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $$200 per ton to process, and ore from source B costs 150 $$per ton to process. Both sources must be kept to less than 50% of the total amount of ore processed. If he uses $$y$$ tons from source B and $$x$$ tons from source A, which system of inequalities represents the constraints?

Accepted Answer

$$x + y \geq 3,\ x \u003c 0.5(x + y),\ y \u003c 0.5(x + y)$$

Question 11

A company makes a product in two different factories. At Factory X it takes 10 hours to produce 100 products and at Factory Y it takes 8 hours to produce 100 products. Each factory can work a maximum of 1200 hours each month. Factory X can produce 6000 products and Factory Y can produce 9000 products each month. If the company must produce at least 12,000 products each month, what is the maximum number of products that can be produced at Factory X and Factory Y combined?

Accepted Answer

$$15,000$$

Question 12

Toy wagons are made to sell at a craft fair. It takes 4 hours to make a small wagon and 6 hours to make a large wagon. The owner of the craft booth will make a profit of $$12 for a small wagon and 30 $$for a large wagon and has no more than 60 hours available to make wagons. The owner wants to have at least 6 small wagons to sell. How many of each size should be made to maximize profit?

Accepted Answer

$$6 	ext{ small wagons and } 6 	ext{ large wagons}$$

Question 13

A company makes a product in two different factories. At Factory X it takes 10 hours to produce 100 products and at Factory Y it takes 8 hours to produce 100 products. Each factory can work a maximum of 1200 hours each month. Factory X can produce 6000 products and Factory Y can produce 9000 products each month. If the company must produce at least 12,000 products each month, what is the maximum number of products that can be produced at Factory X and Factory Y combined?

Accepted Answer

$$15,000$$

Question 14

Suppose a manufacturer makes two types of benches: Home-Fitter and Bomer-Fitter. Each Home-Fitter takes 4 hours to trim and ten hours to sand the lathe, and 2 hours to finish. Each Bomer-Fitter needs 5 hours to trim, 8 to sand, and 6 to finish. The available workers can provide 60 hours for trimming, 100 hours for sanding, and 36 hours for finishing. How many of each type of bench should be produced to make a maximum profit?

Accepted Answer

$$6 	ext{ Home-Fitter and } 3 	ext{ Bomer-Fitter}$$

Precalculus Study Notes: Linear Systems, Word Problems, and Linear Programming

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