Expert: Richard J. Raridon - 10/31/2011

Question
Studying for an algebra 2 final, and I'd like to get the answers to these problems so I can check my work to make sure I got them right. Thanks!

1.   If y is directly proportional to the square root of x, and y = 25 when x = 3, find x when y = 100.
  
x = 3Ö   3
     x = 48
  x = 100      x = 12


2.   If a, b, and c are positive, and a divided by b is equal to b divided by c, then b is called the mean proportion, or geometric mean, between a and c. Find the mean proportion between the pair of numbers: 3 and 27.
  b = 9      b = 81
  b = 27      b = 18


3.   The centrifugal force acting on an object moving in a circle is directly proportional to the square of the speed of the object. If the force is 2240 newtons when the object is moving 8 m/s, what is the force when the object is moving at 12 m/s?
  5,040 n      420 n
  840 n      504 n


4.   If p varies inversely as the square root of q, and p = 12 when q = 36, find p when q = 16.
  p = 4.5      p = 0.056
  p = 18      p = 1.8


5.   The current in an electric circuit of constant voltage varies inversely as the resistance. When the current is 30 amps, the resistance is 20 ohms. Find the current when the resistance is 25 ohms.
  24 amps      22 amps
  20 amps      16 amps


6.   A number y varies inversely as the square of x. If y = 2 when x = 6, what is y when x = 3?
  6      8
  10      12


7.   The height of a cylinder with constant volume is inversely proportional to the square of its radius. If h = 8 when r = 4, what is r when h = 2?
  2      4
  6      8


8.   Find the remainder when t + 1 is divided into

P(t) = t5 + t4 + t3 + t2 + t + 1
  -2      1
  0      3


9.   Find the remainder when z + 2 is divided into

P(z) = z5 + 2z4 + z3 + 2z2 + z + 2
  2      1
  3      0


10.   Find an equation with integral coefficients that has the given roots:

1, -3, -4
  x3 + 12x2 + 5x - 12 = 0      x3 + 6x2 + 5x - 12 = 0
  x3 - 6x2 + 5x - 12 = 0      x3 - 12x2 + 10x - 24 = 0


11.   Determine the missing roots:

Equation:  x4 - 3x3 + 4x2 - 6x + 4 = 0

Roots:   
1, 2, iÖ   2
  
-iÖ   2
     
-iÖ   5
  -i      
iÖ   2


12.   Solve given the root indicated:

Equation:  x4 - 5x2 - 10x - 6 = 0

Root:  -1 + i
  {3, 1, -1, ±i}      {-1 ± i, 3, -1}
  {3, -1, -1 + i}      {-1, -i, i, 1}


13.   Find all roots of the equation

x3 - x2 - 14x + 24 = 0
  -4, 2, 3      4, 2, 3
  0, 4      -4, 3


14.   Find all roots of the equation

6y5 - 13y4 - 6y3 + 17y2 - 4 = 0
  -1, -0.5, 2/3, 1, 2      1, 0.5, 2/3, -1, -2
  -1, 0.5, 1, 2      0, -1, -0.5, 2/3, 1, 2


15.   Find all the zeros of the polynomial

(x + 1)(x - 4)(x2 - 7)
  -1, 4      
-1, 4, ±Ö   7
  No zeros      0


16.   Find all rational roots of:

x5 - 2x4 + 11x3 - 22x2 - 12x + 24 = 0
  1, -1      1, -1, 2
  0, 1, 2      i, -i, 1, 2


17.   Use the Rational Roots Theorem to solve the equation for the rational roots.

4y5 + 8y4 - 29y3 - 42y2 + 45y + 54 = 0
  -3, 2, -1, ± 1.5      -3, 1.5, -1
  2, -1.5, -1, 3      8, -2, 3, 1.5


18.   Find the midpoint of the line segment joining the points:

(13,6) and (0,6)
  (6.5,0)      (13,12)
  (6,6.5)      (6.5,6)


19.   Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(6, -2) and M(0, 5)
  (4, 8)      (6, -12)
  (-6, 12)      (12, -6)


20.   Find the equation of the circle:

center on line x + y = 4; tangent to both first quadrant coordinate axes
  (x + 2)2 + (y + 2)2 = 2      (x - 2)2 + (y - 2)2 = 2
  (x + 2)2 + (y + 2)2 = 4      (x - 2)2 + (y - 2)2 = 4


21.   Find an equation of an ellipse having the given intercepts:

x-intercept: ±3

y-intercept: ±4
  
x2/3    +    y^2/4    =    1
     
x2/9    -    y^2/16    =    1
  
x2/9    +    y^2/16    =    1
  
x2/2    -    y^2/4    =    1



22.   Identify the conic section whose equation is given:

3x2 - 12x + y + 7 = 0
  Parabola      Circle
  Hyperbola      Ellipse


23.   Choose the correct foci:

5x2 + 9y2 = 45
  (±2, 0)      (±4, 0)
  (0, ±2)      (0, ±4)


24.   Choose the correct foci:

25x2 - 144y2 = 3600
  (0, ±13)      (5, 12)
  (12, 5)      (±13, 0)


25.   Solve:

c3 + c2 - 7c - 3 = 0, given root -3
  c = 3 or -1      c = -3, -0.41, 2.41
  c = 1      c = -1


26.   Choose the real solution of the system:

9x2 + 9y2 = 1 and x = y2 + 1
  (2/3, 1/3)      (-5/3, 8/3)
  (0, 0)      No real solution


27.   A 20 m ladder and a 15 m ladder were leaned against a building.  The bottom of the longer ladder was 7 m farther from the building than the bottom of the shorter ladder, but both ladders are the same distance up the building.  Find the distance.
  12m      16m
  14m      10m


28.   A number y varies directly as x and z and inversely as the square of r. If y = 6 when x = 3, z = 4, and r = 7, wheat is y when x = 6, z = 8, and r = 4?
  73.5      60.5
  32.4      84.2


29.   Solve:

(3n-1)-2/3 = 1/4
  3      27
  6      9


30.   
5x = Ö   125
  
2/3
     3
  
1/3
     
3/2


31.   
49x-2 = 7Ö   7
  
-5/4
     
5/4
  
11/4
     
-11/4


32.   Solve the equation:

loga (3x + 5) - loga (x - 5) = loga 8
  9      4.09
  8      3.54


33.   Write the equation in logarithmic form:
e7 = 1097
  ln 1097 = 7      7 ln = 1097
  ln 7 = 1097       1097 ln = 7


34.   Simplify:

eln0.5
  0.5      0.20
  5      0.05


35.   Find a formula for the nth term of the geometric sequence:
500, 100, 20, 4, ...
  500 (1/5)n-1      100 (1/5)n-1
  (1/5)n-1      50 (1/5)n-1


36.   Find a formula for the nth term of the geometric sequence:
8, 12, 18, 27, ...
  8 (3/2)n-1      12n-1
  4 (3/2)n-1      12n-1


37.   Choose a formula for the nth term of the sequence of the odd integers greater than two.
  2n+1       n+1
  2n-1      n-1

        
40.   Find S20 if the series 1+1.1+... is arithmetic.
  2.9      1.9
  39      78


41.   Kirsten is given a test consisting of 15 questions. The first question is worth 5 points, and each question after the first is worth three points more than the question before it. What is the maximum score that Kirsten can obtain?
  390      161
  315       450


42.   Sketch the angle in standard position, indicating its rotation by a curved arrow. Choose the quadrant where the angle is located. Angle: -90°
  IV      II
  Quadrantal angle      I


43.   Express the following as a function of an acute angle: sin (-46.6°)
  -sin 133.4°      -sin 46.6°
  sin 46.6°      sin 133.4°


44.   Choose the correct function value for: sin 26.57°
  0.5527      0.4384
  0.4473      0.5782


45.   Choose the correct function value for: sin 312° 40'
  -0.7353      0.2647
  -0.5492      0.4532


46.   Choose the measure of the acute angle θ, where:

tan θ = 1.325
  53°       127°
  37°      94°


47.   Given oblique triangle ABC, find the length of a: b = 12 c = 17 angle A = 74°
  a = 17.9036       a = 19.2476
  a = 20.3647      a = 16.3246


48.   Given oblique triangle ABC, find the length of c: b = 3 a = 4 angle C = 40°
  c = 8.3910      c = 2.5720
  c = 6.4279      c = 4.3728


49.   Given oblique triangle ABC, find the measure of angle C: a = 9 b = 10 c = 15
  99.9°      104.1°
  56.31°       84.78°


50.   A monument consists of a flagpole 15 m tall standing on a mound in the shape of a cone with vertex angle 140°. How long a shadow does the pole cast on the cone where the angle of elevation from the end of the shadow to the sun is 62°?
  10.52421 m      21.3793 m
  17.8218 m      12.3452 m


51.   Find the area of a rhombus that has a perimeter 48 and an angle of 55°.
  205.6533      82.5950
  94.6753      117.95


52.   Find the area of a regular octagon inscribed in a unit circle (a circle with a radius equal to 1).
  2.8284       4
  3.3190      6


53.   Write an equation for a line through (-3, 5) and (2, 0).
  y = -2 + x      y = x - .5
  y = x + 2       y = -x + 2


54.   Convert the given angle to the appropriate radian value: 300°
  5.24      4.76
  5.82      4.64


55.   A merry-go-round 40 feet in diameter makes four revolutions every minute. What is the speed of the seat on the rim?
  88.57 ft/min      502.65 ft/min
  1,005.31 ft/min      54.36 ft/min


56.   The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 30, bearing 215°
v: magnitude 30, bearing 110°
w: 3u - v
  w: (17, 149°)      w: (102, 231.5°)
  w: (193, 232°)      w: (29, 185°)


57.   The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 136, bearing 220°
v: magnitude 197, bearing 300°
w: u + v
  w: (258.1, 268.7°)      w: (268.7, 258.1°)
  w: (55.2, 223.4°)       w: (223.4, 55.2°)


58.   The magnitude and direction of vectors u and v are given. Find vector w's polar coordinates.
u: magnitude 3.62, bearing 25°
v: magnitude 14.5, bearing 105°
w: u + v
  (15.5, 92.1°)      (92.1, 15.5°)
  (22.4, 19.8°)      (19.8, 22.4°)


59.   Give the polar coordinates: (0, -3).
  (270, 3)      (3, 270)
  (270, 3°)      (3, 270°)


60.   Give the polar coordinates: (-2, 2).
  (2√2, 135°)       (135, 2√2°)
  (2√2, 135)      (135, 2√2)

Answer
I usually don't work that many problems for any one person, but here's most of them.
1. 4(3^1/2)
2. 9
3. 5040
4. 18
5. 24
6. 8
7. 8
8. 0
9. 0
10. x^3+6x^2+5x-12
11. -i(2^1/2)
12. -1 +/-i, 3,-1
13. -4,2,3
14. -1,-0.5,2/3,1,2
15. -1,4, +/-(7^1/2)
18. 6.5,0
19. -6,12
20. (x-2)^2+(y-2)^2 = 4
21. x^2/9 +y^2/16 = 1
22. parabola
23. +/-2,0
24. 0, +/-13
25. c=1
26. no real solution
27. 12m
28. 73.5
29. n=3
30. x=5^1/2
32. x=9
33. ln(1097) = 7
35. 500(1/5)^n-1
36. 8(2/3)^n-1
37. 2n=1
40. 39
41. 390
43. -sin(133.4)
44. 0.4473
45. -0.7353
46. 53 degrees
47. a = 17.9036
48. c = 2.5720
49. 104.1 degrees
53. y = -x+2
54. 5.24
55. 502.65 ft/min