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MTH-5101 Pretest A Solutions

x + y ≥ 25 x + y ≤ 40 x ≥ 25. x ≥ 0 y ≥ 0. MTH-5101 Pretest A Solutions. DIMENSION 1.

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MTH-5101 Pretest A Solutions

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  1. x + y ≥ 25 x + y ≤ 40 x ≥ 25 x ≥ 0 y ≥ 0 MTH-5101 Pretest A Solutions DIMENSION 1 • A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue. • a) Transcribe the elements needed to establish the constraints. • b) Transcribe the elements needed to establish the function to be optimized. • A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue. • a) Transcribe the elements needed to establish the constraints. • b) Transcribe the elements needed to establish the function to be optimized. • A company wants to put a new cleaning product on the market. It has to start with a maximum of 42000 litres of this product that will sell in containers of 500 mL (0.5L) and of 750 mL (0.75L). The requests for containers of 500 mL is at least double that of 750 mL. However the company has decided to produce a minimum of 20000 containers of 750 mL. The price has been fixed at $2.50 for the 500 mL containers and at $3.50 for the 750 mL containers. How many containers of each kind must it sell to maximize revenue. • a) Transcribe the elements needed to establish the constraints. • b) Transcribe the elements needed to establish the function to be optimized. DIMENSION 2 • The community center activities of the municipality wants to publish a guide advertising the sports and cultural activities for the summer. The publisher asks $0.20 for the printing of a page containing photographs and $0.15 for a page without photographs. The guide will have between 25 and 40 pages with a minimum of 10 pages containing photographs. • How many pages will the guide contain for its price to be minimum? • a) Identify the variables. • b) Translate the constraints into a system of inequalities. • c) Express the function to be optimized. Let x = the number of pages with photographs Let y = the number of pages without photographs 0.2x + 0.15y = Z

  2. C3: y = -2x + 16 0 0 x x 8 16 16 y 16 0 y 8 DIMENSION 3 3. Draw the polygon of constraints associated with the following system of inequalities. C1: x ≥ 0 C2: y ≥ 0 C5: y ≥ 2 C3: y ≤ -2x + 16

  3. DIMENSION 3 continued 4. Verify algebraically whether the point (250,100) belongs to the polygon of constraints. Show all steps to your solution. x + y ≤ 300 250 + 100 ≤ 300 350 ≤ 300 False x ≥ 100 250 ≥ 100 250 ≥ 100 True x + y ≥ 180 250 + 100 ≥ 180 350 ≥ 180 True y ≤ x + 60 100 ≤ 250 + 60 100 ≤ 310 True x ≥ 0 250 ≥ 0 True y ≥ 0 100 ≥ 0 True The point (250,100) does not belong to the polygon of constraints because it makes the constraint x + y ≤ 300, false.

  4. 800 600 x + y = 450 x = 2y 2y + y = 450 x + y = 450 y = 0 x + 0 = 450 x = 450 400 3y = 450 y = 150 C x = 2y x = 2(150) x = 300 200 B D A x = 2y 4x + 10y = 3600 4(2y) + 10y = 3600 18y = 3600 y = 200 200 400 600 800 4x + 10y = 3600 y = 0 A x + y = 450 y = 0 (450,0) 4x + 10(0) = 3600 4x = 3600 x = 900 B x + y = 450 x = 2y (300,150) x = 2y x = 2(200) x = 400 C D 4x + 10y = 3600 y = 0 4x + 10y = 3600 x = 2y (400,200) (900,0) DIMENSION 4 5. Determine algebraically the coordinates of the vertices of the polygon of constraints. C1: x ≥ 0 C4: x + y ≥ 450 C2: y ≥ 0 C5: 4x + 10y ≤ 3600 C3: x ≤ 2y

  5. B x + y = 500 A 400 x = 250 y = 0 0 400 x x 0 500 0 y 0 500 100 y x = 4y x + y = 500 4y + y = 500 5y = 500 y = 100 300 y = 0 x + y = 500 x + 0 = 500 x = 500 x = 250 x = 4y 250 = 4y y = 62.5 x = 4y 200 C x = 4y x = 4(100) x = 400 100 100 200 300 400 D DIMENSION 5 6. Alain is the owner of a clothing store. He wishes to sell coats that cost him $25 each and pants at $15 each. He is able to store a maximum of 500 units. He hopes to sell at least 250 pairs of pants and a least four times more pants than coats.. If Alain charges $25 for pants and $45 for coats, how many of each must he sell to maximize his profits? Show all steps of your work. Let x = the number of pairs of pants sold Let y = the number of coats sold 10x + 20y = Z x ≥ 0 x ≥ 250 y ≥ 0 x ≥ 4y x + y ≤ 500

  6. B A 400 300 A x = 250 y = 0 (250,0) 200 B x = 250 x = 4y (250,62.5) C 100 C D x + y = 500 y = 0 x + y = 500 x = 4y (400,100) (500,0) 100 200 300 400 D Let x = the number of pairs of pants sold Let y = the number of coats sold 10x + 20y = Z A(250,0) 10x + 20y = 10(250) + 20(0) = 2500 + 0 = $2500 B(250,62.5) 10x + 20y = 10(250) + 20(62.5) = 2500 + 1250 = $3750 C(400,100) 10x + 20y = 10(400) + 20(100) = 4000 + 2000 = $6000 D(500,0) 10x + 20y = 10(500) + 20(0) = 5000 + 0 = $5000 To maximize profits he must sell 400 pairs of pants and 100 coats.

  7. x + y = 500 x = 30 y = 50 150 0 x 0 150 y B x = 30 x + y = 150 30+ y = 150 y = 120 y = 50 x + y = 150 x + 50 = 150 x = 100 A x = 30 y = 50 (30,50) B x + y = 150 x =30 (30,120) A C C x + y = 150 y = 50 (100,50) 7. In a hunting and fishing camp, we offer rifle hunting and bow hunting excursions. In order to preserve the wildlife, excursions are carried out according to certain constraints. There is a maximum of 150 registrants There must be at least 30 registrants for rifle hunting and 50 registrants for bow hunting. If $500 is charged for each rifle hunting excursion and $400 for each bow hunting excursion, how many excursions for rifle hunting and bow hunting are necessary to maximize revenue? Let x = the number of rifle hunting excursions Let y = the number of bow hunting excursions x ≥ 0 500x + 400y = Z x ≥ 30 y ≥ 0 y ≥ 50 x + y ≤ 150 200 150 100 50 50 100 150 200

  8. 200 150 100 50 B 50 100 150 200 A x = 30 y = 50 (30,50) B x + y = 150 x =30 (30,120) A C C x + y = 150 y = 50 (100,50) Let x = the number of pairs of pants sold Let y = the number of coats sold 500x + 400y = Z A(30,50) 500x + 400y = 500(30) + 400(50) = 15000 + 20000 = $35000 B(30,120) 500x + 400y = 500(30) + 400(120) = 15000 + 48000 = $63000 C(100,50) 500x + 400y = 500(100) + 400(50) = 50000 + 20000 = $70000 100 rifle excursions and 50 bow excursions are necessary to maximize revenue.

  9. 0 x 10 5 0 y DIMENSION 6 • Alex wants to buy himself some pants and sweaters. The saleswoman tells him that a pair of pants costs $50 and a sweater costs $25. His father offers to pay for his purchases as a gift for his admission to CEGEP. • The polygon of constraints is presented in the graph on the right where: 8 x = the number of sweaters y = the number of pants 6 Alex wants at least 2 times more sweaters than pants. What will be the impact of this additional consideration on the minimal cost that his father will have to pay? (4,4) 4 Function of Optimization: 25x + 50y = Z (6,3) (2,2) 25x + 50y = 25(2) + 50(2) = 50 + 100 = $150 2 (2,2) (5,2) (5,2) 25x + 50y = 25(5) + 50(2) = 125 + 100 = $225 (6,3) 25x + 50y = 25(6) + 50(3) = 150 + 150 = $300 (4,4) 25x + 50y = 25(4) + 50(4) = 100 + 300 = $300 2 4 6 8 According to the polygon of constraints presented, the minimal cost to pay would be $150. With the additional consideration, the minimal cost would have to be higher because with 2 sweaters and 2 pants Alex will NOT have 2 times more sweaters than pants. x ≥ 2y (4,2) 25x + 50y = 25(4) + 50(2) = 100 + 100 = $200 To be consistent with the additional consideration, the father would have to buy 4 sweaters and 2 pants at a cost of $200.

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