022test2reviewf071 by HC120914062145

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									9/07


                                                                                           MATH 022
                                                                                             TEST 2
                                                                                      REVIEW SHEET
TO THE STUDENT:

This Review Sheet gives you an outline of the topics covered on Test 2 as well as practice
problems.

NEW TOPICS:
A. Solving Quadratic Equations.
For problems 1 – 16, find the solutions for each of the following equations by taking the square
root or by factoring.
                                                          9.   9 x 2  36
       1.      2 x  23x  6  0                                  3    x 1
                                                              10.        
       2.     x  2 x  15  0
                2
                                                                     x4     2
              8 x
       3.      
              x 9                                             11.    x  5 x  5  0
                                                              12.   16 x 2  64 x  64  0
       4.      8 x 2  72  0

              x 2  8 x  15  0
                                                              13. 2a 3  18a  0
       5.
                                                                        64
              x 2  16 x  0                                  14. x 2 
       6.                                                               81
       7.     x 2  36  12 x                                15.   3x  25x  4  0

       8.    121x 2  64  0                                  16.   49  y 2


For problems 17 – 20, a) solve each of the following square root equations.
                                b) check each solution.


       17.          3 x  x 3                               19.     x 1  1 x

       18.          x4  x2                                 20.     x2  x2


       21. An air-powered rocket is launched, and at time t it is at the height h(t )  16t 2  64t
           where t is in seconds and height is in ft.
           a) Find the time in seconds where it would reach the maximum height.
           b) Find the maximum height.
22. Suppose a connector in a high-pressure water system has broken and a fine spray of water
    is shooting straight up into the air with the height h(t )  16t 2  48t where h is height in
    feet and t is time in second.
    a) In how many seconds after leaving the break will a droplet of water reach the 32 feet
        level?
    b) How long it will take a droplet of water come back and reach the ground?

23. A rocket in a fireworks display is shot from ground level. Its height in feet with time in
    seconds is h(t )  16t 2  96t .
    a) Find the height of the rocket after 3 seconds.
    b) When the rocket will reach at the height of 140 ft?
    c) How long it will take the rocket to reach its maximum height?
    d) How long it will take the rocket to come back down to ground?

For problems 24 – 31, use graph or table to solve each of the following quadratic equations.

24.       x2  4 x  8  3                            28.    x 2  5x  8  6
25.       x 2  2 x  3  3                          29.    t 2  8t  40  8
26.       2  x  x2  4                              30.    x 2  8 x  16
27.       m2  11m  33  9                           31.    x 2  5x  8  15


32. In what situation does a graph of a quadratic function opens upward?

For problems 33 – 38, find a) x-intercepts
                          b) y-intercept
                          c) axis of symmetry
                          d) vertex

33.       x 2  6 x  16  0                          36.   x 2  13x  42  0
34.       x 2  12 x  20  0                         37.   x 2  16 x  64  0
                                                      38.   18 x 2  3x  6  0
35.       x 2  6 x  27  0


                                               1
For problems 39–40, assume the formula h  h0  gt 2 holds true. Let g = 32.2 ft per second
                                               2
squared.

39. Suppose a drop of water starts at the top of the American side of Niagara Falls and falls
    182 feet.
      a) How long will it take to reach halfway?
      b) How long will it take the drop to reach the bottom?



                                              2
   40. a) Write the equation and find the time a sugar cube dropped from a height of 500 feet
          will be at 200 feet.
       b) How long will it take to reach the bottom?

B. Solving Square Root Equations.

   41. a) Graph y  4  x and y = 3. Copy these two graphs on your paper.
         b) For what input, x, is 4  x defined?
         c) Solve 4  x  3 from the graph. Make sure to mark the point and write down the
            solution.

   42. Use a graphing calculator to find the solution of x  3  2
       a) Copy the graph on your paper.
       b) Mark the solution on the graph paper and write down the solution.
       c) Use algebra to solve the equation.



   For the following equations, a) find inputs which will be defined.
                                b) solve each of the following equations.


   43.        2x 1  7                                45.        6  2x  4

   44.        x  3  1                               46.        x3  2

                                                                                           3h
   47. On the moon, the distance seen in miles from a height of h feet is given by d         .
                                                                                            8
         How high do we need to be on the moon to see 60 miles?


C. Square Root Expressions and Simplify Expressions.

   48. For 9  x , evaluate and simplify
       a) f (–3)
       b) f (0)
       c) f (12)

   49. For what inputs that     3x will be defined?


   50. Simplify each of the following expressions where each variable represent any real
       number.


                                               3
        a)        x6 y8                                   f)
                                                                     28 y
                                                                     7 y3
        b)        a2b4                                                2 xy
                                                          g)
                                                                     50 x 5 y 3
        c)      400x 2                                   h)         32
                  3x 4                                    i)         32a         2a
        d)
                  27                                                      6 y3
                                                                 
               6 2 
                           2                              j)
        e)                                                                24 y

D. Exponents.

   51. Simplify each of the following expressions. All the answers should have all positiv
       exponents.


                x 3 y 2                                         8 x 4 y 2
        a)                                                c)
                 xy 3                                           10 x 3 y 4
                          1                                                3
                a                                              2y 
        b)                                              d)     2
                bc                                            x 


   52. Simplify each of the following expressions. Leave answers without denominators.


        a)
               a 5                                       d)    y   2 3

               a3                                                1
        b)     x 5  3                                 e)
                                                                 x4
        c)     n 7 n 2                                   f)    x y 2   3 4



   53. Change 32,794,810 kilometers to scientific notation. Round the decimal part to two
       decimal places.

   54. Change the scientific notation 5.39  104 kilograms to decimal notation.

   55. If a silver birch tree is 30 feet tall after 15 years of growth, how fast in feet per second is
       growing? Make sure the answer is in scientific notation with unit. Round the decimal part
       to two decimal places.

                           1 year = 365 days
                           1 day = 24 hours


                                                  4
OLD TOPICS:

  56. Use graphing calculator to graph 4  2 x  3(2 x  4) .
      a) Copy the graph on a graph paper. Label each axis and x-min, x-max,
          y-min and y-max.
      b) Use algebraic to solve this inequality.
      c) Graph the solution set on a number line.
      d) Express the answer in interval notation.

  For problems 57 – 60, multiply and simplify each of the following expressions.
         2 x(3x 2  4 x  2)                                  a  4
                                                                         2
  57.                                                   59.
  58.    5  4 x 5  4 x                            60.   2  4  2 x  x2   x 4  2 x  x2 

  For problems 61 – 66, factor each of the following expressions.

  61.  x 2 y  xy 2  y 3                              64.   2a 3  18a
  62. 24 x 2 y  6 xy 2                                 65.   4t 2  12t  9
  63. 9a 2  36a  36                                   66.   3t 2  18t  15

  For problems 67–68, solve each problem using an algebraic approach. You may need a
  quantity-value table or you may need a diagram. You may wish to set up a system of two
  equations in two variables or you may wish to set up one equation in one variable. Use
  whatever tool is appropriate for the problem. Make sure to define each variable you use.

  67. An English muffin and two fried eggs contain 330 calories. Three English muffins and
      one fried egg contain 515 calories. How many calories are in one English muffin and one
      fried egg?

  68. Five grams of carbohydrate and 2 grams of fat contain 38 calories. Two grams of
      carbohydrate and 6 grams of fat contain 62 calories. How many calories are in 1 gram of
      carbohydrate and 1 gram of fat?

  For problems 69 – 70, solve the system of equations by any method.

           2 x  3 y  7                                       5 x  2 y  36
  69.                                                  70.     
           3 y  4 x  1                                      25 x  7 y  51


  71. A line goes through the points (–10, –1) and (10, 1). Write the equation of this line.

  72. Do the following equations represent two parallel lines? Explain clearly.

                2 x  2 y  100      y  20  x


                                                  5
                                                                                       MATH 022
                                  ANSWERS TO PROBLEMS                                    TEST 2
                                                                                  REVIEW SHEET


NEW TOPICS:

A. Solving Quadratic Equations.                      24.    x  1.87, x  5.87
                                                     25.   x = 0, x = –2
   1.    x = 1, x = –2                               26.   no solution
   2.    x = –5, x = 3                               27.   m = 8, m = 3
   3.     x  6 2 , x  6 2                         28.   x = 7, x = –2
                                                     29.    t = –12, t = 4
   4.    x =3, x = –3                                30.   x = 4,
   5.    x = –3, x = –5
   6.    x = 0, x = 16
                                                     31. no solution
   7.    x = –6                                      32. When the leading coefficient, a > 0
   8.    x = 8/11, x = –8/11                         33. a) x = 8, x = –2
                                                         b) b = –16
   9.    x = 2, x = –2
                                                         c) x = 3
   10.   x = 5, x = –2
   11.   x = 5, x = –5                                   d) (3, –25)
   12.   x=2                                         34. a) x = 10, x = 2
   13.   a =0, a =3, a = –3                              b) 20
                                                         c) x = 6
   14.   x = 8/9, x = –8/9
                                                         d) (6, –16)
   15.   x = 2/3, x = 4/5
   16.   x = 7, x = –7                               35. a) x = 9, x = –3
   17.   a) x = 3, x = 2                                 b) –27
              x3               x2                      d) x = 3
                                                         d) (3, –36)
         b)    3 3  3 3        3 2  2  3
          0  0                             36. a) x = 6, x = 7
          x = 3 is the correct answer                    b) 42
   18. a) x = 0, x = 5                                   c) x = 6.5
           x0                x5                        d) (6.5, –0.25)

         b)     04  02          5 4  5 2       37. a)     x=8
              2  2             33                   b)     64
              x = 5 is the correct answer                c)     x=8
                                                         d)     (8, 0)
   19. a) x = 2, x = 1
                                                     38. a)     x = 2/3, x = –1/2
       b) check. x = 1 is the correct answer
                                                         b)     –6
                                                         c)     x = 1/12 or 0.083
   20. a) x = 3, x = 2
       b) check. x = 2, x = 1 are both correct           d)     (0.083, –6.125)
          answers
   21. a) t = 2 sec.
                                                                           1
       b) height is 64 ft.                           39. a)     91  182  (32.2)t 2 ; t  2.38sec
                                                                           2
   22. a) when t = 1 sec. And t = 2 sec.                                  1
                                                           b)   0  182  (32.2)t 2 ; t  3.36 sec
       b) when t = 3 sec.                                                 2
                                                                             1
   23. a)     144 ft.                                40. a)     200  500  (32.2)t 2 ; t  4.5 sec
                                                                             2
       b)     t = 3.5 sec, x = 2.5 sec.                                    1
       c)     t = 3 sec                                    b)   0  500  (32.2)t 2 ; t  5.5 sec
       d)     t = 6 sec                                                    2




                                                 6
B. Solving Square Root Equations.                                         46. a) x  3
                                                                              b) x = 7
       41. a) , c)                                                        47. 9600 ft.

                                                                       C. Square Root Expressions and Simplify
                                          y                               Expressions.
                                      
     (-5,3)                  y = 3                                        48. a) 2 3
                                                                              b) 3
                                          y = sqrt(4-x)                       c) not a real number
                                      
                                                                          49. when x  0
                                                                          50. a) x 3 y 4
                                                          x                   b)   a b2
                                              
                                                                              c)   20 a
                                                                                    x2
                                                                              d)
                                                                                  3
                                                                              e)   72
                                                                                    4
                                                                              f)
                                                                                    y
                                                                                     1
b) x  4                                                                      g)
                                                                                   5x 2 y
42. a)                                                                        h)   4 2
              y

                                                                              i)   8a
       
                                                                                         y
                                                                              j)   
                                          y = sqrt(x-3)                                  2
       
                                                                       D. Exponents.
                                                                                   y5
                                                                          51. a)
                                                                                   x4
                                                                  x                bc
                                                                              b)
                                                                             a
                                                                                   4x
                                                                              c)
                                                                                   5y2
      
                       y = -2                                                       x6
                                                                              d)
                                                                                   8 y3
                                                                          52. a)   a 8
                                                                              b)   a15
       b) no intersection, no solution                                        c)   n 5
           c) show work
                                                                              d)   y6
                   1
       43. a) x                                                              e)   x 4
                   2
           b) x = 25                                                          f)   x8 y12
                                                                          53. 3.28 107
       44. a) x  3                                                       54. 0.000539
           b) no solution
                                                                          55. 6.34 108
       45. a) x  3
           b) x = –5



                                                              7
OLD TOPICS:

   56. a) and c)




         b) ( –, 2]

   57. 6x3  8x2  4 x
   58. 25  16x2
   59. a2  8a  16
   60. x3  8
   61.  y  x 2  xy  y 2 
   62. 6 xy  4 x  y 
   63. 9  a  2 
                      2


   64. 2a(a  3)(a  3)
          2t  3 
                  2
   65.
   66.    3t  3 t  5

   67. Let m = the number of calories in one
       English muffin
           e = the number of calories in one egg.
       m + 2e = 330
       3m + e = 515
       m = 140, e = 95
       There are 140 calories in one English muffin
       and 95 calories in one fried egg.
   68. Let c = the number of calories in one gram of
       carbohydrates
           f = the number of calories in one gram of
       fat.
       5c + 2f = 38
       2c + 6f = 62
       c = 4, f = 9
       There are 4 calories in one gram of
       carbohydrates and 9 calories in one gram of
       fat.

           5
   69. 1,  
           3
   70. (–10, 43)
            1
   71. y     x
           10
   72. Yes, same slope.




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