FOM 11 Practice Test Name: Ch. 6 Systems of Inequalities Block:

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1 FOM 11 Practice Test Name: Ch. 6 Systems of Inequalities Date: Block: Multiple Choice Identify the choice that best completes the statement or answers the question. 1. For which inequality is ( 5, 1) a possible solution? a. y > 9 b. y 2x 10 c. y 9 + 2x d. y < x 2 2. What is the boundary line for the linear inequality y 2x 10? a. y 2x 10 b. y 2x + 10 c. y 2x 10 d. y 2x What system of linear inequalities is shown here? a. 2x + 3y 6 y > 2x 3 b. 2x + 3y < 6 y > 2x 3 c. 2x + 3y < 6 y 2x 3 d. 2x + 3y? 6 y 2x 3

2 4. Describe the boundary lines for the following system of linear inequalities. {y 3x < 12, x + y 0, x R, y R} a. Dashed line along y = 3x + 12; solid line along y = x b. Dashed line along y = 3x + 12; dashed line along y = x c. Solid line along y = 3x + 12; dashed line along y = x d. Solid line along y = 3x + 12; solid line along y = x 5. A vending machine sells juice and pop. The machine holds, at most, 200 cans of drinks. Sales from the vending machine show that at least 3 cans of juice are sold for each can of pop. Each can of juice sells for $1.50, and each can of pop sells for $1.00. Let x represent the number of cans of pop. Let y represent the number of cans of juice. How would you write the objective function for revenue, R? a. R = x y b. R = 1.25x + y c. R = 1.50(x + y) d. R = 1.50y x 6. A vending machine sells juice and pop. The machine holds, at most, 200 cans of drinks. Sales from the vending machine show that at least 3 cans of juice are sold for each can of pop. Each can of juice sells for $1.50, and each can of pop sells for $1.00. Let x represent the number of cans of pop. Let y represent the number of cans of juice. Which of the following is a constraint of this optimization problem? a. 3x y b. x 3y c. x 3y d. 3x y 7. Brent found spiders and grasshoppers in his barn. There were at most 12 spiders and at least 10 grasshoppers. There were no more than 36 spiders and grasshoppers, in total. Let s represent the number of spiders and let g represent the number of grasshoppers. Which inequality represents a restriction of s and g based on the given information? a. s + g > 36 b. s g 36 c. s g 22 d. s + g 36

3 Short Answer 8. Graph the solution set for the linear inequality 5y 2x The following model represents an optimization problem. Determine the coordinate that will result in a maximum solution (of the objective function). Restrictions: x W y W Constraints: x > 0 y > 0 5x y + 5 x + y 19 Objective function: A = x + 2y

4 10. A butcher shop makes hamburger patties and sausages. Hamburger patties sell for $2 and sausage sell for $1.50. The butcher noticed that they always sell at least twice as many sausages as hamburger patties The butcher never sells more than 100 hamburger patties or 300 sausages. Let h represent the number of hamburger patties sold. Let s represent the number of sausages sold. a) Write a system of linear inequalities to describe the constraints. b) Write an objective function that represents the profit made from the sale of hamburger patties and sausages. 11. A cafeteria offers pepperoni and vegetarian pizza slices. Pepperoni slices sell for $3 and vegetarian slices sell for $2.50. Every day they sell at least three times as many pepperoni slices as vegetarian slices. They always sell at least 40 slices of vegetarian pizza. The total sales are never more than 240 slices. What are the maximum and minimum profits for a month? Let x be the number of pepperoni slices sold and y be the number of vegetarian slices sold

5 Problem 12. Gordon s favourite activities are going to the movies and skating with friends. He budgets himself no more than $180 a month for entertainment and transportation. Movie admission is $12 per movie, and skating costs $10 each time. A student bus pass for the month costs $50. (Hint: this means his budget for movies and skating is $50 less) a) Define the variables and write a linear inequality to represent the situation. b) Graph the linear inequality. c) Use your graph to determine i) a combination of activities that Gordon can afford with no money left over ii) a combination of activities that will exceed his budget

6 13. A pet store specializes in birds. It sells at least four times more male birds than female birds of the same species. (The males colourful feathers make them more popular.) Over the past month, no more than 40 birds, in total, have been sold. Males were sold for $120, and females were sold for $75. a) Create a model of this problem. (Define Variables, and List Constraints) b) What is the feasible region of sales of male and female birds?

7 14. On a flight between Calgary and Thunder Bay, there are business class and economy seats. At capacity, the airplane can hold no more than 133 passengers. No fewer than 124 economy seats are sold, and no more than 5 business class seats are sold. The airline charges $624 for business class seats and $239 for economy seats. What combination of business class and economy seats will result in the maximum revenue? What will this maximum revenue be?

8 FOM 11 Practice Test Ch. 6 Systems of Inequalities Answer Section 11. y y 40 y x MULTIPLE CHOICE 1. B 2. B 3. A 4. A 5. A 6. A 7. D (180,60) (120,40 (200,40) SHORT ANSWER y Objective Function: P = 3x + 2.5y Maximum: $700 (200, 40) Minimum: $460 (120, 40) x y 5x 5 and y x + 19 (4, 19) PROBLEM 12. a) Let x = # movies Gordon sees. Let y = # times Gordon goes skating. (x W, y W) (0, 13) 12x + 10y b) Graph the line 13 (1,0) (19,0) Coordinate that gives the maximum solution in the Objective Function is (4, 19) 10. Constraints: h 0 (implied by saying that h W) s 0 (implied by saying that s W) h 100 s 300 2h s Objective function: P = 2h + 1.5s (12, 1) i) (anything along the boundary line) eg. 6 movies and go skating 7 times 12(5) + 10(7) + 50 = 180 ii) (anything not in shaded region) e.g., see 7 movies and go skating 7 times 12(7) + 10(7) + 50 = 204 > 180

9 13. a) Let x = # male birds sold. Let y = # female birds sold. (x W, y W) 4y x x + y 40 y = x + 40 Objective function to maximize: R = 120x + 75y The feasible region is all the whole number points in the shaded area and its boundaries. (32, 8) (40, 0) 14. Let x = # of economy seats. Let y = # of business class seats. (x W, y W) Constraints: x + y 133 x 124 y 5 Use technology to graph the lines and find the intersection points of the solution area. Intersection points are: (124, 0), (133, 0), (124, 5) and (128, 5) Objective function to maximize: R = 239x + 624y Feasible Region! The maximum is at point (128, 5), which represents 128 economy seats and 5 business class seats. R = 239(128) + 624(5) R = The maximum revenue is $