Precalc warm ups


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Precalc warm ups

  1. 1. Honors Precalc Warm-Ups
  2. 2. Use a calculator to approximate the following: What does this expression represent?
  3. 3. If r is a real number, either positive, negative, or zero, which of the following is always greater than r? Why?
  4. 4. Six students each try to guess thenumber of pennies in a jar. The sixguesses are 52, 59, 62, 65, 49, and42. One guess is 12 away and theother guesses are 1, 4, 6, 9, and 11 away. How many pennies are in the jar?
  5. 5. Graph the function (without a calculator): f(x) = | x + 1 | + | x – 1|
  6. 6. Each of the circles has a 10 inch radius. ABCD is a square inscribed in a semi-circle; MNPQ is a square inscribed in acircle. What is the ratio of the area of ABCD to MNPQ?
  7. 7. If a circle and a square have the same perimeter, which of the following statements are true? A. Their areas are equal.B. The area of the circle is greater than the area of the square. C. The area of the square is greater than the area of the circle.D. The area of the circle is π times the area of the square.
  8. 8. Draw an example of an octagon with: A. One line of symmetry B. Two lines of symmetry C. Four lines of symmetry D. Eight lines of symmetry
  9. 9. A vertical line divides the trianglewith vertices (0,0), (1,1) and (9,1) into two regions of equal area.What is the equation of the vertical line?
  10. 10. The radius of a cylinder is 8 inches. The height of the cylinder is 3 inches. What number of inches can be added to either the radius or the height to give the same positive increase in volume?
  11. 11. A circle with area 9π is inscribed inan equilateral triangle. What is the area of the triangle?
  12. 12. Let ΔABC be a right triangle with AC as hypotenuse. Let CAB be acircle of diameter AB. Let CBC be acircle of diameter BC. Let CAC be a circle of diameter AC. Show that area CAB + area CBC = area CAC
  13. 13. What is the 500th fraction in the sequence?What is the n th fraction in the sequence?
  14. 14. The point (3, 5) is the top vertex of an isosceles triangle with baseparallel to the x-axis. The triangle has an area of 12. Find the coordinates of the other two vertices.
  15. 15. If the circumference of a pipe is increased from 20 inches to 25inches, what is the increase in the radius?
  16. 16. An 8 x 4 rectangle is centered at(-3, 7). What are the coordinates of the four corner points of the rectangle is the longer side is parallel to the y-axis?
  17. 17. In this figure, MNOP is a squareand DAN is an equilateral triangle.If the area of MNOP is one square inch, what is the area of DAN in square inches?
  18. 18. Given a regular octagon with sides of length 8, find the distance between opposite sides.
  19. 19. Given f(x) =ax 2+ bx + 5. Find a and b such that f(x + 1) – f(x) = 8x + 3
  20. 20. A wheel has an outside diameter of 6 feet. How many revolutions of the wheel are required to cause a point on the rim to go one mile?
  21. 21. Two cars start together and move in the same direction around a circular race track of length 999meters. The cars meet for the first time 37 minutes after they start.Determine the speed of each car ifthe speed of the first car is 4 times the speed of the second.
  22. 22. A rectangular box measures 4 feet by 4 feet by 3 feet. Will a 6-foot pole fit inside? Justify your answer!
  23. 23. Solve the problems below. Eachletter represents a unique digit. Letter values differ in the two problems.
  24. 24. The graph of a parabola that is symmetric about a vertical linecontains the points (1, 4), (3, 18), and (0, 3). Find the equation of the parabola.
  25. 25. Describe and draw the solid that results from revolving a right triangle about one of its legs.
  26. 26. The odometer of the family car showed 15,951 miles. The driver noticed that this number was palindromic, that is, it read the sameforward or backward. Two hours later, the odometer showed a newpalindromic number. How fast was the car traveling in those two hours?
  27. 27. The function f(x) = x3 - 6x2 + 9xhas a relative extremum at x = 3. Is it a maximum or a minimum?
  28. 28. ABCD and EFGH are squares, each with a side of length 12. E is thecenter of ABCD. If the distance BJ is 3, find the area of EJCK.
  29. 29. A fox notices a rabbit that is 30 meters away. The fox’s leap is 2meters whereas the rabbit’s leap is1 meter. If the fox can take 3 leapsin the time that the rabbit can take 2 leaps, what distance will the fox have to cover to catch the rabbit?
  30. 30. Find the points of intersection of the two graphs given by the equations 4x 2 + y2 = 100 and 9x2 – y2 = 108
  31. 31. Which is larger? or
  32. 32. n and k are integers. What is the smallest value for n such thatn 3 – n is not divisible by k for all 1 ≤ k ≤ n?
  33. 33. Find the volume of a spherecircumscribed about a 4 cm by 4 cm by 4 cm cube.
  34. 34. Color the vertices of the given figurewith as many colors as needed such that no two adjacent vertices arecolored the same color. What is theminimum number of distinct colors needed?
  35. 35. If (2137)753 is multiplied out, theones digit in the final product will be ________.
  36. 36. If the length of the diagonal of asquare is y + w, what is the area of the square?
  37. 37. Three circles with centers (-2, 3), (0, 4), and (4, 6) are all tangent toeach other at the point (2, 5). Find an equation for each circle.
  38. 38. An equivalent expression for is:
  39. 39. The medians of a triangle meet atthe centroid. Find the centroid of triangle ABC if A is (1, -2), B is (-2, 4), and C is (1, 6)
  40. 40. A tennis ball is served from aheight of 7 feet to just clear a net 3 feet high. If it is served from a line 39 feet behind the net and travelsin a straight path, how far from the net does the ball first hit the ground?
  41. 41. For what values of x will the graphof y = x 2 – x – 2 be below the graph of y = x + 1?
  42. 42. The dot product of two vectors isdefined as (a1, a2, a3 )(b1, b2, b3 ) = a1 b1 + a2 b2 + a3 b3. Find the dot product of (7, 6, 3) and (2, -1, 6).
  43. 43. Suppose a cable fits tightly around the equator of the earth. If an additional piece is to be spliced into the cable so that it can be raised three meters above the earth at all points, how much additional cable will be needed?
  44. 44. The altitudes of a triangle are concurrent at the orthocenter. Find the orthocenter of triangleABC if A is (-2, 2), B is (4, 3), and C is (3, -2)
  45. 45. If f(x) is the inverse of g(x) underfunction composition, what is therelation of the graph of f(x) to the graph of g(x)?
  46. 46. If [a] denotes the greatest integerless than or equal to a, what is the value of [a] + [-a] when a is a real number?
  47. 47. Solve for x.sin 2x = 2 cos x
  48. 48. When an ice cube melts until its sides are one third their originallength, what fractional part of the ice cube has melted away?
  49. 49. Let A(-3, 1), B(2, 3), and C(-2, 5) be three points. If line CX is perpendicular to line AB at thepoint X, find the coordinates of X.
  50. 50. How far apart are the two circles x2 + y2 + 4x – 12y + 31 = 0 and x 2 + y2 - 10x – 2y + 10 = 0?
  51. 51. What are the x- and y- intercepts of the hyperbola given by the equation
  52. 52. The current in an electrical circuitfollows the equation l = 5(sin 8.72t)where t is the time in seconds and l is the current in amperes. What is the maximum value of the current in this circuit?
  53. 53. How does cos x vary as x goes from π to ?
  54. 54. If a > 1 and 0 < x < y, which is larger, loga x or loga y?
  55. 55. Two high school classes took the same math test. One class of 20students made an average grade of 80%. The other class had an average grade of 70%. If the combined average was 74%, howmany students were in the second class?
  56. 56. If f(x) is any periodic function with period p, thenA. m · f(x) has period __________B. f(x + m) has period __________C. f(mx) has period ___________
  57. 57. A regular hexagon is inscribed in acircle of radius 10 inches. What is the area of the hexagon?
  58. 58. The location of a cannon is given coordinates (0, 100). The cannonfires a projectile which follows the path .for x ≥ 0. What are the coordinates of the point where the projectile hits the positive x-axis?
  59. 59. A parachutist will land somewhere in the square. What is theprobability that she will land in the shaded area?
  60. 60. A box contains 5 red marbles and 3green marbles. Three marbles are selected one after the otherwithout replacing them as they are selected. What is the probability that the third marble selected is green?
  61. 61. A kite escaped the hands of a child.The little boy watched as the kite rose 200 m, traveled 1000 m due northwest, dropped 100 m, and continued to travel 500 m northeast.Then it turned and traveled 1000 m tothe southeast and finally dropped 100 m. How far was the kite from its starting place?
  62. 62. Simplify the continued fraction:
  63. 63. Solve for x.log4 (x – 5) + log4 (x + 1) = 2
  64. 64. A circular race track has a 50-meter radius for one running lane and a 52-meter radius for the second running lane. Two runners are beginning a race. How much of a headstart should the outside runner be given?
  65. 65. How many pairs (a, b) of non-zeroreal numbers satisfy the equation ?
  66. 66. Solve for x.
  67. 67. On a square piece of paper, drawthe largest possible circle and cut it out. Inside the circle, draw the largest possible square and cut it out. The final square is what fraction of the original piece of paper?
  68. 68. Four girls bought a bike for $60. Thefirst girl paid one half of the sum of the amounts paid by the other girls; the second girl paid one third of the sumof the amounts paid by the other girls; the third girl paid one fourth of the sum of the amounts paid by the other girls. How much money did the fourth girl pay?
  69. 69. What is the area of the trianglelying between the coordinate axes and the line 3x + 8y – 24 = 0?
  70. 70. A ladder is 16 feet long. The top of the ladder rests against a window of a two-story house that is builton level ground. The ladder makes an angle between 700 and 850 with the ground. What are the minimum and maximum heights from the ground to the window?