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MTH-4111 PRETEST. 1. Solve the following system of equations:. 5 marks. Students. f. m. 1980. 2000. Year.

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## 1. Solve the following system of equations:

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**MTH-4111 PRETEST**1. Solve the following system of equations: 5 marks**Students**f m 1980 2000 Year In 1980, the student population of a CEGEP was 220 girls and 570 guys. Over the next 20 years, the number of girls increased steadily to 480. The population of guys followed a quadratic relationship and reached their maximum of 810 students. If these tendency continues, how many female students will there be when there are as many girls as guys? What year should this happen? 10 marks Equation of m: y = a(x - h)2 + k y = a(x - 20)2 + 810 570 = a(0 – 20)2 + 810 570 – 810 = 400a 400a = -240 a = -0.6 Slope of f: 13x + 220 = -0.6x2 + 24x + 570 0.6x2 - 11x – 350 = 0 x = -16.67 or x = 35 y = -0.6(x - 20)2 + 810 y = -0.6(x2 – 40x + 400) + 810 y = -0.6x2 + 24x – 240 + 810 y = -0.6x2 + 24x + 570 Equation of f: y = 13x + 220 y = 13(35) + 220 y = 455 + 220 y = 675 In the year 2015, the number of girls will be equal to the number of guys each at 675.**A)**C) B) D) 3. Two functions are described below. f(x) = bwhere b < 0 g(x) = ax2 + cwhere a > 0 and c = - b. 5 marks Which of the following graphs represents the function operation, g – f? Assume b = -1 and therefore c = 1. g – f = ax2 +1 – (-1) = ax2 + 2 • Orientation will remain positive as a > 0 still (parabola opens up) • y-int will be made higher for g – f than for g • These characteristics are only displayed in A.**h**D) C) g A) 5 marks • Functions g and h are represented graphically to the right. • Identify the graph below that corresponds to g • h. B) (Function g is negative) х (Function h with a positive slope) Product will give a linear function with a negative slope. The only choice that displays this characteristic is B.**B(5, 6)**C(4, -2) A(-8, -5) • The points A (-8, -5); B (5, 6) and C (4, -2) determine the vertices of triangle ABC. • Express the equation of the median from point A. 5 marks**A (-3, 4)**B (3, 2) D (-1, -2) C (5, -4) C (4, 7) B (-4, 5) A (-8, -3) D (-2, -5) 6. Find the area of the rhombus illustrated in the diagram. Round your answer to the nearest whole number. 10 marks 7. The vertices of a quadrilateral are as follows: A (-8, -3), B (-4, 5), C (4, 7) and D (-2, -5). Prove that this quadrilateral is a trapezoid. The slopes of the bases are equal which means they are parallel and ABCD is a trapezoid. 10 marks**y**HYPOTHESIS: B (a,c) CONCLUSION: D E x C (2a,0) A (0,0) STATEMENTS JUSTIFICATIONS 8. Complete the demonstration of following proposition using geometric analysis. THE MEDIANS TO THE CONGRUENT SIDES OF AN ISOSCELES TRIANGLE ARE ALSO CONGRUENT. 5 marks 1. The coordinates of D and E are: 1. Midpoint Formula: We can therefore say that the 2 medians are congruent.**C**B CONCLUSION: O A D STATEMENTS JUSTIFICATIONS 9. Complete the demonstration of following proposition: THE DIAGONALS OF A PARALLELOGRAM INTERSECT AT THEIR MIDPOINTS. 5 marks HYPOTHESIS: (b,c) (b+a,c) 1. Midpoint Formula: (0,0) (a,0) The coordinates of the midpoints for both diagonals are the same so they do indeed intersect at their midpoints.**B**C 40° A D E x B x+4 40° A D C E D • ΔABD and ΔECD are similar and the ratios of their areas is 0.36. • The measure of angle A is 40° and segment BC measures 4 cm. Determine the perimeter of ΔABD. 10 marks Perimeter = 10 + 11.92 + 15.56 = 37.48 cm**E**F B G D A C • ΔABC is equivalent to trapezoid DEFG. • The base of the triangle measures 20 cm more than its height. • In the trapezoid, the short base is two-thirds of the height. The length of the long base is equal to the sum of the lengths of the short base and the height. The sum of the two bases and the height of the trapezoid is 160 cm. • Determine the measures of the base and height of the triangle. 10 marks TRAPEZOID Let x = height = 48 cm = 32 cm = 80 cm Area of ΔABC =Area of trapezoid DEFG = 2688 cm2 TRIANGLE Height = 64 cm and Base = 84 cm.**3 cm**9 cm • Two cylindrical blocks of cheese are similar. The wrapping of one that has a diameter of 9 cm is 2.25 times more than the wrapping of the other that has a height of 3 cm. What is the difference between the volumes to the nearest tenth of a cubic centimeter between the two cheese portions? Cylinder 2 h2= 3 cm d2 = ? r2 = ? Cylinder 1 h1 = ? d1 = 9 cm r1 = 4.5 cm Cylinder 2 V2= πr22h2 =(3.142)(3)2(3) =84.82 cm3 Cylinder 1 V1= πr12h1 = (3.142)(4.5)2(4.5) = 286.28 cm3 Difference in volumes = V1 – V2 = 286.28 – 84.82 = 201.5 cm3 10 marks**70º**2.4 m 70° 2.4 m • A funnel is used to fill cubic boxes with fuel for an expedition. Their capacities are equivalent to each other. • The slant height of the cone is 2.4 meters and it forms a 70° with the radius of the base of the cone. What is the measure, to the nearest tenth, of the side of the box? r h Cone and Cube are equivalent. Vcube = Vcone Vcube= 1.591 cm3 Vcube= s3 s3= 1.591 10 marks

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