Mathematician Chapter 8 Review Answers

Chapter 8 Review ANSWERS Mathematician ____________________________________ 1. Mr. Lowery, former Olympian for the USA Hockey Team, produces ice skates. He can make as many as 60 pairs of hockey skates and 45 pairs of figure skates per day. It takes 3 hours of labor to produce a pair of hockey skates and 4 hours of labor to produce a pair of figure skates. His company has up to 240 hours of labor to produce ice skates each day. If the profit is $25 on each pair of hockey skates and $35 for each pair of figure skates, then find the number of pairs of each kind of skate the firm should produce to gain the maximum profit each day. Write the objective function Write the constraints (there are 3) P = 25x + 35y P – profit X – pairs of hockey skates Y – pairs of figure skates 3. Graph the inequalities. Make sure to label the graph.  (0, 45)  (20, 45) Solve for: 3x + 4y = 240 by substitution y = 45 3x + 4(45) = 240 3x = 60 x = 20  (60, 15)  (0, 0)  Solve for: 3x + 4y = 240 by substitution x = 60 3(60) + 4y = 240 4y = 60 y = 15 (60, 0) 4. Find the points of intersection. (0,0) (0, 45) (20, 45) (60,15) (60, 0) 5. What is the maximum profit? P = 25x + 35y = 25(0) + 35(0) =0 = 25(0) + 35(45) = 1575 = 25(20) + 35(45) = 2075 = 25(60) + 35(15) = 2025 = 25(60) + 35(0) = 1500 The maximum profit is $2,075. To obtain Mr. Lowery should produce 20 hockey skates and 45 figure skates. Coach Castillo, PHS track coach, decides to open his own business by making shot puts and discuses. Each shot put requires 5 hours of sculpting and 4 hours of finishing. Each discuss requires 10 hours of sculpting and 3 hours of finishing. Each month his business has 600 hours available for sculpting and 240 hours for finishing. The profit on each shot put is $40 and on each discuss is $75. How many of each product should be make each month in order to maximize profit? Write the objective function Write the constraints (there are 3) P = 40x + 75y P – profit X – number of shot puts produced Y – number of discuses produced 3. Graph the inequalities. Make sure to label the graph. 4. Find the points of intersection. (0, 0) (0, 60) (24, 48) (60, 0)  (0, 60)  (24, 48) Solve by elimination: -3(5x + 10y = 600 ) 10(4x + 3y = 240 ) -15x -30y = -1800 40x + 30y = 2400 25x = 600 x = 24 5(24) + 10y = 600 10y = 480 y = 48 5. What is the maximum profit? P = 40x + 75y = 40 (0) + 75 (0) = 0 = 40 (0) + 75 (60) = 4,500 = 40 (24) + 75 (48) = 4,560 = 40 (60) + 75 (0) = 2,400 The maximum profit of $4,560 can be achieved by making 24 shot puts and 48 discuses.  (0, 0)  (60, 0) y = 4x + 3 -20x + 4y = -8 y = 3x – 2 15x – 5y = 10 x = -2y 3x + 4y = -2 15x – 5(3x – 2) = 10 15x-15x+10 = 10 10 = 10 -20x + 4(4x + 3) = -8 -20x + 16x +12 = -8 -4x = -20 x =5 y = 4(5) +3 = 20+3 = 23 Infinitely Many Solutions 3(-2y) + 4y = -2 -6y + 4y = -2 -2y = -2 y=1 x=-2(1) = -2 (5, 23) (Note: If these were rewritten in the slopeintercept format, y=mx+b, they would be the same line) (-2, 1) 4x + 6y = 12 6x + 9y = 12 8x + 6y = 4 6x – 3y = -42 3x + 2y = 48 9x – 8y = -24 3(4x + 6y = 12) -2(6x + 9y = 12) 8x + 6y = 4 2(6x – 3y = -42) 4(3x + 2y = 48) 9x – 8y = -24 12x +18y = 36 -12x -18y = -24 0 = 12 8x + 6y = 4 12x – 6y = -84 20x = -80 x = -4 8(-4) +6y = 4 6y = 36 y=6 12x + 8y = 192 9x – 8y = -24 21x = 168 x=8 No Solution (Note, if written in the slope-intercept form, y=mx+b, you can see that both lines have the same slope and are, therefore, parallel.) 3(8) + 2y = 48 2y = 24 Y = 12 (-4, 6) (8, 12)