This document contains 20 math word problems covering a variety of topics including: geometry (area of shapes, properties of circles, triangles, etc.), algebra (equations, functions, graphs), trigonometry, probability, and more. The problems range from basic calculations to more complex multi-step problems.

1. Honors Precalc Warm-Ups

2. Use a calculator to approximate the following:
What does this expression represent?

3. If r is a real number, either positive,
negative, or zero, which of the
following is always greater than r?
Why?

4. Six students each try to guess the
number of pennies in a jar. The six
guesses are 52, 59, 62, 65, 49, and
42. One guess is 12 away and the
other guesses are 1, 4, 6, 9, and 11
away. How many pennies are in
the jar?

5. Graph the function (without a
calculator):
f(x) = | x + 1 | + | x – 1|

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 a
circle. What is the ratio of the area of
ABCD to MNPQ?

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. 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. A vertical line divides the triangle
with vertices (0,0), (1,1) and (9,1)
into two regions of equal area.
What is the equation of the vertical
line?

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. A circle with area 9π is inscribed in
an equilateral triangle. What is the
area of the triangle?

12. Let ΔABC be a right triangle with
AC as hypotenuse. Let CAB be a
circle of diameter AB. Let CBC be a
circle of diameter BC. Let CAC be a
circle of diameter AC.
Show that
area CAB + area CBC = area CAC

13. What is the 500th fraction in the
sequence?
What is the n th fraction in the
sequence?

14. The point (3, 5) is the top vertex of
an isosceles triangle with base
parallel to the x-axis. The triangle
has an area of 12. Find the
coordinates of the other two
vertices.

15. If the circumference of a pipe is
increased from 20 inches to 25
inches, what is the increase in the
radius?

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. In this figure, MNOP is a square
and DAN is an equilateral triangle.
If the area of MNOP is one square
inch, what is the area of DAN in
square inches?

18. Given a regular octagon with sides
of length 8, find the distance
between opposite sides.

19. Given f(x) =ax 2+ bx + 5. Find a and
b such that f(x + 1) – f(x) = 8x + 3

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. Two cars start together and move
in the same direction around a
circular race track of length 999
meters. The cars meet for the first
time 37 minutes after they start.
Determine the speed of each car if
the speed of the first car is 4 times
the speed of the second.

22. A rectangular box measures 4 feet
by 4 feet by 3 feet. Will a 6-foot
pole fit inside?
Justify your answer!

23. Solve the problems below. Each
letter represents a unique digit.
Letter values differ in the two
problems.

24. The graph of a parabola that is
symmetric about a vertical line
contains the points (1, 4), (3, 18),
and (0, 3). Find the equation of
the parabola.

25. Describe and draw the solid that
results from revolving a right
triangle about one of its legs.

26. The odometer of the family car
showed 15,951 miles. The driver
noticed that this number was
palindromic, that is, it read the same
forward or backward. Two hours later,
the odometer showed a new
palindromic number. How fast was the
car traveling in those two hours?

27. The function f(x) = x3 - 6x2 + 9x
has a relative extremum at x = 3. Is
it a maximum or a minimum?

28. ABCD and EFGH are squares, each
with a side of length 12. E is the
center of ABCD. If the distance BJ
is 3, find the area of EJCK.

29. A fox notices a rabbit that is 30
meters away. The fox’s leap is 2
meters whereas the rabbit’s leap is
1 meter. If the fox can take 3 leaps
in the time that the rabbit can take
2 leaps, what distance will the fox
have to cover to catch the rabbit?

30. Find the points of intersection of
the two graphs given by the
equations 4x 2 + y2 = 100 and
9x2 – y2 = 108

31. Which is larger?
or

32. n and k are integers. What is the
smallest value for n such that
n 3 – n is not divisible by k for all 1 ≤
k ≤ n?

33. Find the volume of a sphere
circumscribed about a 4 cm by 4
cm by 4 cm cube.

34. Color the vertices of the given figure
with as many colors as needed such
that no two adjacent vertices are
colored the same color. What is the
minimum number of distinct colors
needed?

35. If (2137)753 is multiplied out, the
ones digit in the final product will
be ________.

36. If the length of the diagonal of a
square is y + w, what is the area of
the square?

37. Three circles with centers (-2, 3),
(0, 4), and (4, 6) are all tangent to
each other at the point (2, 5). Find
an equation for each circle.

38. An equivalent expression for
is:

39. The medians of a triangle meet at
the centroid. Find the centroid of
triangle ABC if A is (1, -2), B is
(-2, 4), and C is (1, 6)

40. A tennis ball is served from a
height of 7 feet to just clear a net 3
feet high. If it is served from a line
39 feet behind the net and travels
in a straight path, how far from the
net does the ball first hit the
ground?

41. For what values of x will the graph
of y = x 2 – x – 2 be below the graph
of y = x + 1?

42. The dot product of two vectors is
defined 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. 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. The altitudes of a triangle are
concurrent at the orthocenter.
Find the orthocenter of triangle
ABC if A is (-2, 2), B is (4, 3), and C
is (3, -2)

45. If f(x) is the inverse of g(x) under
function composition, what is the
relation of the graph of f(x) to the
graph of g(x)?

46. If [a] denotes the greatest integer
less than or equal to a, what is the
value of [a] + [-a] when a is a real
number?

47. Solve for x.
sin 2x = 2 cos x

48. When an ice cube melts until its
sides are one third their original
length, what fractional part of the
ice cube has melted away?

49. Let A(-3, 1), B(2, 3), and C(-2, 5) be
three points. If line CX is
perpendicular to line AB at the
point X, find the coordinates of X.

50. How far apart are the two circles
x2 + y2 + 4x – 12y + 31 = 0 and
x 2 + y2 - 10x – 2y + 10 = 0?

51. What are the x- and y- intercepts
of the hyperbola given by the
equation

52. The current in an electrical circuit
follows 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. How does cos x vary as x goes from
π to ?

54. If a > 1 and 0 < x < y, which is
larger, loga x or loga y?

55. Two high school classes took the
same math test. One class of 20
students made an average grade of
80%. The other class had an
average grade of 70%. If the
combined average was 74%, how
many students were in the second
class?

56. If f(x) is any periodic function with
period p, then
A. m · f(x) has period __________
B. f(x + m) has period __________
C. f(mx) has period ___________

57. A regular hexagon is inscribed in a
circle of radius 10 inches. What is
the area of the hexagon?

58. The location of a cannon is given
coordinates (0, 100). The cannon
fires 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. A parachutist will land somewhere
in the square. What is the
probability that she will land in the
shaded area?

60. A box contains 5 red marbles and 3
green marbles. Three marbles are
selected one after the other
without replacing them as they are
selected. What is the probability
that the third marble selected is
green?

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 to
the southeast and finally dropped 100
m. How far was the kite from its
starting place?

62. Simplify the continued fraction:

63. Solve for x.
log4 (x – 5) + log4 (x + 1) = 2

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. How many pairs (a, b) of non-zero
real numbers satisfy the equation
?

66. Solve for x.

67. On a square piece of paper, draw
the 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. Four girls bought a bike for $60. The
first girl paid one half of the sum of the
amounts paid by the other girls; the
second girl paid one third of the sum
of 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. What is the area of the triangle
lying between the coordinate axes
and the line 3x + 8y – 24 = 0?

70. A ladder is 16 feet long. The top of
the ladder rests against a window
of a two-story house that is built
on 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?