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MATH 1010 Sec. 4 Solution Midterm 1 October 21, 2010 (Total: 100 points + extra credits: 5 points) You have to show all the work on the space given. Please be neat! If you need more space, you can use extra sheets of paper provided. (Please put your name on all the sheets and label question numbers.) Not showing your work gets 0 point. Where appropriate, clearly indicate your final answer by circling it. You are not allowed to get help from your textbook, class notes, calculators, other students, or any other form of outside aid. Do not forget to turn off your cell phone. If you have any question, please ask the instructor. Good luck! 1. (Total: 10 points) Solve the following equations (a) (5 points) 5x − 4 + 2(2x − 3) = 12x − 3(5x + 1) (b) (5 points) |5x − 9| = 3x + 1 (a) 5x − 4 + 2(2x − 3) = 12x − 3(5x + 1) 5x − 4 + 4x − 6 = 12x − 15x − 3 9x − 10 = −3x − 3 9x + 3x = 10 − 3 12x = 7 7 x= 12 (b) |5x − 9| = 3x + 1 We have two possibilities: 5x − 9 = 3x + 1 or 5x − 9 = −3x − 1 In the first case we get x = 5. In the second case we get x = 1. Therefore, the solutions are x = 5 and x = 1. 1 2. (10 points) You are driving from Rome to Paris, a trip of 1000 kilometers. Your average speed is 50 kilometers per hour. How long does it take you to reach your destination? Let time = t. Since speed = we have distance time 1000 t 1000 = 20 t= 50 It takes 20 hours to go from Rome to Paris. 50 = 2 3. (Total: 10 points) Solve the following linear inequalities (a) (5 points) 3x − 4 > 2 (b) (5 points) 4x + 5 ≤ x + 2 (a) 3x − 4 > 2 3x > 6 x>2 The set of solutions is {x | x > 2}. (b) 4x + 5 ≤ x + 2 4x − x ≤ 2 − 5 3x ≤ −3 x ≤ −1 The set of solutions is {x | x ≤ −1}. 3 4. (5 points) The number 6 is 30% of what number? Let n be a number 6 = 30% × n 6 6 n= = = 20 30% 0.3 5. (10 points) Determine whether the set of points {A = (0, 4), B = (4, 1), C = (3, 0)} is collinear. Explain why. The distance between A and B is p √ √ d1 = (4 − 0)2 + (1 − 4)2 = 16 + 9 = 25 = 5 The distance between A and C is p √ √ d2 = (3 − 0)2 + (0 − 4)2 = 9 + 16 = 25 = 5 The distance between B and C is p √ √ d3 = (4 − 3)2 + (1 − 0)2 = 1 + 1 = 2 Since d1 6= d2 + d3 , the points are not collinear. 4 6. (5 points) Rewrite 6, 300, 000 into scientific notation. 6, 300, 000 = 6.3 × 106 7. (5 points) Simplify 6x2 y 3x5 y −3 2y 1+3 2y 4 6x2 y = = 3x5 y −3 x5−2 x3 5 8. (10 points) Sketch the graph of the line given by 8x + 4y = 3. Then find the slope of the line, the x-intercept, and the y-intercept. The line is y = −2x + 43 . Slope: −2. x-intercept: ( 38 , 0). y-intercept: (0, 43 ). y x 6 9. (Total: 10 points) Let f (x) = x2 − 3x − 1. Find each value of the function. (a) f (−2) = (−2)2 − 3(−2) − 1 = 4 + 6 − 1 = 9 (b) f (3) + f (0) = 32 − 3(3) − 1 + (−1) = 9 − 9 − 1 − 1 = −2 10. (10 points) Solve the following system of equations −x + 2y = 5 2x − 3y = 1 From the first equation x = 2y − 5. Substituting it in the second equation we have 2(2y − 5) − 3y = 1 4y − 10 − 3y = 1 Therefore, y = 11. Substituting the value of y in x = 2y − 5 we get x = 2(11) − 5 = 22 − 5 = 17 The solution is (17, 11). 7 11. (10 points) Solve the following system of linear equations x − 3y + z = 6 3y + 2z = 5 3y − z = 2 Do R3 − R2 in the third row and get x − 3y + z = 6 3y + 2z = 5 −3z = −3 Divide by −3 the third row x − 3y + z = 6 3y + 2z = 5 z = 1 Now substituting in the second row we get y = 1. Substituting y = 1, z = 1 in the first equation we find x = 8. The solution is (8, 1, 1). 8 12. (10 points) Sketch the graph of the following x−y x y The lines are y = x − 5, x = 3, and y = 0. y y=0 x=3 x y=x−5 9 system of linear inequalities < 5 > 3 ≤ 0 13. (Extra problem: 5 points) An object is dropped from the top of the Tower of Pisa, which is 180 feet high. The initial velocity is 0 feet per second. Use the position function h(t) = −16t2 + v0 t + s0 to find the time when the object hits the ground. (You can leave the solution as a fraction). We have h(t) = −16t2 + 180 = 0 16t2 = 180 180 t2 = 16 r 180 t= 16 The object hits the ground in q 180 16 seconds. 10