# 0.5

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## 0.5

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1. m 5 4 3 2,3 2 ? 1 0.5 2.5 m The graphs to the right illustrate a fountain shooting a jet of water with a laser beam illuminating the jet in two places, one of them being (2 m ,3 m). Determine the height of the other point illuminated by the laser. Equation of the jet of water x1 = 0.5; x1 = 2.5; (x,y) = (2,3) y = a(x – x1)(x – x2) 3 = a(2 – 0.5)( 2 – 2.5) 3 = a(1.5)(-0.5) 3 = -0.75a a = -4 y = -4(x – 0.5)(x – 2.5) y = -4(x2 – 2.5x – 0.5x + 1.25) y = -4x2 + 12x – 5 Solution of the system x + 1 = -4x2 + 12x – 5 4x2 - 12x + x + 5 + 1 = 0 4x2 - 11x + 6 = 0 4x2 - 3x - 8x + 6 = 0 x(4x – 3) – 2(4x – 3) = 0 (4x – 3)(x – 2) = 0 4x – 3 = 0 OR x – 2 = 0 Equation of the laser beam The height of the other point illuminated by water is 1.75 meters.

2. men women # of employees year The proportion of men to women employed by a company has varied over the years of operation. The number of men was 428 in the year 2000 and reached a maximum of 500 three years later following a second degree curve. The number of women in 2000 was 100 and increased at a rate of 25 per years after that. If the trends continue, in what year will the number of men and women be the same? How many will be employed by the company at that time? x: number of years after 2000 y: number of employees Equation for men Vertex: (3,500); Other point: (0,428) y = a(x – h)2 + k 428 = a(0 – 3)2 + 500 428 – 500 = 9a 9a = -72 a = -8 Equation for women y = -8(x – 3)2 + 500 y = -8(x2 – 6x + 9)+ 500 y = -8x2 + 48x - 72 + 500 y = -8x2 + 48x + 428 m = 25; b = 100 y = 25x + 100 Solution of the System 25x + 100 = -8x2 + 48x + 428 8x2 - 48x + 25x – 428 + 100 = 0 8x2 - 23x – 328 = 0 8x2 - 64x + 41x – 328 = 0 8x(x – 8) + 41(x – 8) = 0 (x – 8)(8x + 41) = 0 x – 8 = 0 OR 8x + 41 = 0 x = 8 OR x = -5.125 After 8 years, there were 300 men and 300 women employed for a total of 600 employees y = 25x + 100 y = 25(8) + 100 = 200 + 100 = 300 employees

3. GASPE Time Temp 4 8 12 16 20 24 -2.4°C 0.4°C 2.4°C 3.6°C 4°C 3.6°C A cold front sweeps into the town of Amos causing the temperatures to drop steadily from 1°C at 2h to -4°C at 21h and continue. As yet unaffected by this cold front, on the same day Gaspe experiences temperatures indicated in the table to the right. For how long is the temperature of Gaspe warmer than that of Amos during that time? Equation for Amos x = time y = temperature Solution of the System Equation for Gaspe Vertex: (20,4); Other point: (8, 0.4) y = a(x – h)2 + k 0.4 = a(8 – 20)2 + 4 0.4 – 4 = 144a 144a = -3.6 a = -0.025 y = -0.025(x – 20)2 + 4 y = -0.025(x2 – 40x + 400) + 4 y = -0.025x2 + x - 10 + 4 y = -0.025x2 + x - 6 0.475x2 - 19x – 5x + 114 + 29 = 0 0.475x2 - 24x + 143 = 0 a = 0.475; b = -24; c = 143 Δ = b2 – 4ac = (-24)2 – 4(0.475)(143) = 476 – 271.7 =304.3 Gaspe was warmer than Amos between 6.91 hours and 43.6 hours: 43.6 – 6.9 = 36.1 hours.

4. y braces ? seat 2.25 m 1.5 m x 42cm A display is constructed to take pictures of kids with Santa Claus. Two symmetrical braces extend up from the floor to support the seat and attach to a piece of board above the seat. The display is 2.25 meters high and 1.5 meters wide at its base. The braces are 42 cm apart at the floor and 58 cm apart at the seat which is 48 cm above the floor. How long are the braces? How far apart are the braces at the top where they attach to the piece of board? Equation of the brace Solution of the System Equation of the parabola 6x – 1.26 = -4x2 + 2.25 4x2 + 6x - 3.51 = 0 a = 4; b = 6; c = -3.51 Δ = b2 – 4ac = 62 – 4(4)(-3.51) = 36 + 56.16 = 92.16 Vertex: (0, 2.25); Other point: (0.75, 0) y = a(x – h)2 + k 0 = a(0.75 – 0)2 + 2.25 0 – 2.25 = 0.5625a 0.5625a = -2.25 a = -4 y = -4x2 + 2.25 y = 6x – 1.26 = 6(0.45) – 1.26 = 2.7 – 1.26 = 1.44

5. y y (0.45,1.44) (0.45,1.44) braces braces ? ? seat seat 2.25 m 2.25 m 1.5 m 1.5 m x x (0.21,0) 42cm 42cm (-0.45,1.44) End points of right brace: (0.21,0); (0.45,1.44) Length of right brace: (x1, y1) = (0.21,0); (x2, y2) = (0.45,1.44) End points of right brace: (-0.45,1.44); (0.45,1.44) Length of right brace: =|0.45-(-0.45)| =0.9 m

6. \$ expenses revenue month A new business is starting and the manager is analyzing his expenses and revenues. The monthly expenses start at \$1950 and increase to a maximum of \$2550 after 5 months at which point the expenses start to decline. The revenues increase at a constant rate of \$30 per month and reach \$2256 after 12 months. At what point will there be a zero deficit? ( i.e. revenue = expenses) Equation for revenue Equation for expenses m = 30; ( x1 , y1) = (12,2256) Vertex: (5,2550); Other point: (0,1950) y = a(x – h)2 + k 1950 = a(0 – 5)2 + 2550 1950 – 2550 = 25a 25a = -600 a = -24 y = -24(x – 5)2 + 2550 y = -24(x2 - 10x + 25) + 2550 y = -24x2 + 240x - 600 + 2550 y = -24x2 + 240x + 1950 Solution of the System 30x + 1896 = -24x2 + 240x + 1950 24x2 - 240x + 30x – 1950 + 1896 = 0 24x2 - 210x – 54 = 0 4x2 - 35x – 9 = 0 4x2 + 1x - 36x – 9 = 0 x(4x + 1) – 9(4x + 1) = 0 (4x + 1)(x – 9) = 0 4x + 1 = 0 OR x - 9 = 0 x = -0.25 OR x = 9 There is a zero deficit after 9 months.

7. N River Bridges Highway (9,4.2) (5,1) E 1 unit = 1 km Plans for a highway indicate that it will cross a river in 2 places. Use the coordinates from the graph provided to determine the length of the section of the highway to be built between the bridges. Equation for Highway Equation for River Vertex: (5,1); Other point: (9,4.2) y = a(x – h)2 + k 4.2 = a(9 – 5)2 + 1 4.2 - 1 = 16a 16a = 3.2 a = 0.2 y = 0.2(x – 5)2 + 1 y =0.2(x2 – 10x + 25) + 1 y = 0.2x2 – 2x + 5 + 1 y = 0.2x2 – 2x + 6 Solution of the System 0.2x2 – 2x + 6 = 0.4x + 0.6 0.2x2 – 2x – 0.4x + 11 – 0.6 = 0 0.2x2 – 2.4x + 5.4 = 0 x2 – 12x + 27 = 0 (x – 3)(x – 9) = 0 x – 3 = 0 OR x – 9 = 0 x = 3 OR x = 9 y = 0.4(3) + 0.6 = 1.2 + 0.6 = 1.8 (3, 1.8) y = 0.4(9) + 0.6 = 3.6 + 0.6 = 4.2 (9, 4.2)