March 25, 2008
2008 State Math Contest
Wake Technical Community College
Geometry Test
1. The fifth-degree polynomial function p( x ) = a5 x 5 + a 4 x 4 + a3 x 3 + a 2 x 2 + a1 x + a 0 is graphed below:
1000
500
0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-500
-1000
-1500
y
If p (− 5) = 900 , p (− 2 ) = −192 , and p (3) = −972 which of the following most closely approximates the
area between the curve p ( x ) and the x-axis.
a. 3486 sq units
b. 3096 sq units
c. 3000 sq units
d. 3464 sq units
e. 3582 sq units
2. Airport runways are labeled by two numbers less than 36. Airport officials measure the clockwise
angles of the runway’s direction from North to the nearest 10° and divide by 10. If a runway with
heading 206° is labeled 21, what is the runway’s other number?
a. 2
b. 3
c. 15
d. 16
e. 9
3. A ladder leans against a house with its base 15 feet from the house. When the ladder is pulled 9 feet
farther from the house the upper end of the ladder slides 13 feet down. How long is the ladder?
a. 32 feet
b. 7 feet
c. 12 feet
1
d. 24 feet
e. 25 feet
2008 State Math Contest
Wake Technical Community College
Geometry Test
4. In square ABCD, A and B lie on a circle of radius 10 inches, and the circle is tangent to side CD at the
midpoint of CD . What is the area of the square?
a. 144 sq inches
b. 196 sq inches
c. 256 sq inches
d. 324 sq inches
e. 400 sq inches
5. A rectangular swimming pool is two and a half times as long as it is wide and has an 8-foot wide
concrete border around it. If the border has an area of 1936 square feet, what is the perimeter of the
pool?
a. 225 feet
b. 210 feet
c. 190 feet
d. 204 feet
e. 215 feet
6. A newspaper recently reported the following findings:
35% of all residents own a dog.
25% of all residents own a cat.
45% of all residents own at least one pet.
Which of the following statements must be true, based on these findings?
(i) Most residents don’t own a pet.
(ii) 60% of all residents own a cat or a dog.
(iii) 15% of all residents own both a cat and a dog.
a. Only (i) is true.
b. Only (ii) is true.
d. All the statements are true.
c. Both (i) and (ii) are true.
e. None of the statements are true.
7. A company employs two types of hourly workers: skilled and unskilled. Using the information in the
table below, compute the average salary per hour for hourly workers at the company.
Type
Skilled
Unskilled
a. $12.00
b. $10.33
Hourly Workers
Number
Hourly Pay
10
$12.00
20
$8.00
c. $10.00
2
d. $9.33
e. $8.67
2008 State Math Contest
Wake Technical Community College
Geometry Test
8. A company is conscious about the amount of water it uses. There are 24 employees who manage 86
procedures that use a great deal of water. They are asked to document consumption on a daily basis.
One procedure uses water at a rate of 325 gallons per hour. If the procedure started at 11:33 a.m. and
was completed at 3:21 p.m. later that day, about how many gallons of water was used in the procedure?
a. 910 gallons
b. 1131 gallons
c. 1235 gallons
d. 1265 gallons
e. 1560 gallons
9. A recent study suggested that a typical family of
four spends 18% of their income on food
purchased at the grocery store and 12% of their
income on food purchased at a restaurant. The
study also published the pie chart to the right to
show how the money was spent at the grocery
store According to this study if a family of four
has an income of $4250 per month, then to the
nearest dollar how much do they spend on meat at
the grocery store in one month?
a. $219
b. $329
c. $548
d. $765
e. $1827
10. A pizza store owner figures the cost of producing a six-inch (diameter) pizza pie should be $4.99 based
on his costs for materials, labor, and overhead. He wants to base the cost of larger pizzas on this price.
His two larger size pizzas will be eight-inch and 12-inch (diameter). How much more (to the nearest
dime) should the 12-inch pizza cost than the eight-inch?
a. $2.50
b. $3.30
c. $8.70
d. $11.10
e. $20.00
11. In rectangle ABCD, point E is on side AB so that AE = 10 and EB = 5. What fraction of the area of the
rectangle is inside triangle AEC?
a.
1
2
b.
1
3
c.
3
1
4
d.
2
3
e.
2
5
2008 State Math Contest
Wake Technical Community College
Geometry Test
12. In triangle ABC, AB = AC = 25 and BC = 14. The perpendicular distances from a point P in the interior
of triangle ABC to each of the three sides are equal. Find this distance.
a.
9
2
b.
19
4
c. 5
13. An 8x8 checkerboard is exactly covered by 16
of tiles in which the
is horizontal?
a. 0
b. 2
d.
21
4
e.
11
2
shaped tiles. What is the least possible number
c. 4
d. 6
e. 8
14. Cy is standing 12 ft from an office building. Looking directly at the building, he sees Al on the second
floor at an angle of 60° above the horizontal, and Bob on the fourth floor at an angle of 75° above the
horizontal. What is the vertical distance to the nearest foot between Al and Bob?
a. 18 feet
b. 20 feet
c. 22 feet
d. 24 feet
e. 26 feet
15. Al and Bo are each either a knight, who always tells the truth; a knave, who always lies; or an alternator,
who always lies and tells the truth in strict alternation. Al says “I am an alternator.” Bo says, “Al is a
knave.” Al then says, “Bo is not an alternator.” Which of the following must be true?
a. Al is a knave. b. Al is an alternator. c. Bo is an alternator. d. Bo is a knight. e. Bo is a knave.
16. A trapezoid has three sides of length 4 inches and one side of length 8 inches. What is its area?
a. 12 2 sq inches
b. 15 2 sq inches
c. 16 3 sq inches
4
d. 12 3 sq inches e. 21 sq inches
2008 State Math Contest
Wake Technical Community College
Geometry Test
17. The volumes of two cubes differ by 259 cu meters. If the edges of one cube are each 4 meters greater
than the edges of the other, what is the sum of the lengths of one edge of each cube?
a. 8.5 m
b. 9 m
c. 9.5 m
d. 10 m
e. 10.5 m
18. A polyhedron is formed by connecting the midpoints of adjacent edges of a cube. How many faces does
the polyhedron have?
a. 8
b. 10
c. 12
d. 14
e. 16
19. Triangle PQR is equilateral with QR = 30 feet. A is the foot of the perpendicular from Q to PR and B is
the midpoint of QA . What is the length of PB to the nearest hundredth of a foot?
Q
B
P
R
A
a. 19.84 ft
b. 20.03 ft
c. 19.26 ft
d. 20.46 ft
e. 19.56 ft
20. A rectangular solid’s length is increased by 20%, its width is increased by 30%, and its height is
decreased by 40%. What is the percentage change in the volume?
a. 6.4% decrease
b. 4% decrease
c. 2.4 % increase
d. 10% increase
e. 2.8% decrease
21. An arch is in the shape of a semicircle. At a point along the base 1 foot from an end of the arch, the
height of the arch is 7 ft. What is the maximum height of the arch?
a. 21 feet
b. 25 feet
c. 49 feet
5
d. 18 2 feet
e. 16 2 feet
2008 State Math Contest
Wake Technical Community College
Geometry Test
22. Circle A has radius 1 unit and is located in the first quadrant. It is tangent to both the x-axis and the
y-axis. A larger circle B is also located in the first quadrant, is also tangent to both the x-axis and the
y-axis, and is tangent to circle A. What is the radius of circle B?
a. 3 + 2 2 units
b.
(
) units
2 3 +1
3 −1
c. 2 + 2 2 units
(
)
d. 3 2 2 − 1 units
e. 4 3 − 1 units
23. Suppose the earth were a perfect sphere with a perfectly fitting belt of length 24,000 miles surrounding it
along a great circular path. Suppose the belt was cut, and one hundred feet of additional material was
added to the belt, with the “loose fit” evenly distributed around the earth so that the new belt was still
circular with its center at the center of the earth. Which of the following best describes the resulting
situation?
a.
b.
c.
d.
e.
You could slip a piece of paper between the belt and the earth.
You could get your fingers under the belt.
You could crawl under the belt.
You could walk upright under the belt.
You could drive a tractor trailer under the belt.
24. In a regular pentagon with side length one yard the distance from its center to a vertex is approximately
0.85 yards. What is the area of a regular pentagon with a side length of one yard to the nearest
hundredth of a square yard?
a. 1.48 sq yard
b. 1.54 sq yard
c. 1.60 sq yard
d. 1.64 sq yard
e. 1.72 sq yard
25. In the continental United States there are 4 states whose boundaries intersect at right angles meeting at
one point. Designate the intersection point as the origin of a coordinate system with the y-axis oriented
South/North and the x-axis oriented East/West along the boundaries of the states. Which of the
following states lies in more than one quadrant?
a. Arizona
b. Utah
c. Wyoming
6
d. Colorado
e. New Mexico
2008 State Math Contest
Wake Technical Community College
Geometry Test
SHORT ANSWER
Place the answer in the appropriate space.
66. Al and Curt live at opposite corners of a rectangular lot; Beth and Dawn live at the other two corners.
They all carry water from a spring located within the lot. The spring is 50 yards from Al, 30 yards from
Beth, and 40 yards from Curt. How far to the nearest yard is the spring from Dawn?
67. How many rectangular solids are possible with a volume of 200 cubic meters and dimensions of integral
value?
68. Triangle ABC is an isosceles right triangle with BC=AB=3 units. Circular arcs of radius 3 units centered
at C and A meet the hypotenuse at D and E, respectively. What is the area of the shaded region to the
nearest hundredth of a square unit?
A
D
E
B
C
69. A company is decreasing by 8% the amount of soup sold in cylindrical cans. The cans will be the same
height but have a smaller diameter. To the nearest tenth of a percent how much should the diameter be
decreased to accommodate the change?
70. A triangle has sides lengths 29 cm, 35 cm, and 48 cm. To the nearest whole number of centimeters,
what is the perpendicular distance from the side of length 48 cm to the opposite vertex?
7
2008 State Math Contest
Wake Technical Community College
Geometry Test
1. e
2. b
3. e
4. c
5. b
6. a
7. d
8. c
9. b
10. d
11. b
12. d
13. c
14. d
15. c
16. d
17. b
18. d
19. a
20. a
21. b
22. a
23. e
24. e
25. c
66. 57 yd
67. 12
68. 1.93 sq units
69. 4.1%
70. 21 cm
8
2008 State Math Contest
Wake Technical Community College
Geometry Test
1. Approximate all areas with triangles and you get 3582 square units.
2. The other heading for this runway is 206° − 180° = 26° ≈ 30° so it should be labeled 3.
3. Let the ladder’s length be x and the original height on the house be y then use the Pythagorean Theorem
to find y equal to 20 feet. Hence, x equals 25 feet.
4. Let the side of the square be 2x and the center of the circle be O. Construct the perpendicular to side
BC from the center O intersecting the side at point E. Right triangle OBE has sides length 10 inches, 2x
− 10 inches, and x inches, respectively. Use the Pythagorean Theorem to find x equal to 8 inches.
Hence, the area is 256 sq inches.
5. Let the width of the pool be x, then the area of the border satisfies 4(8)(8) + 2(8)(2.5 x ) + 2(8)x = 1936 .
This gives a width of 30 feet and a length of 75 feet.
6. Only (i) is true since 100% − 45% = 55%. There is not enough information to determine whether the
other statements are true.
7.
10(12) + 20(8)
= 9.33
30
8.
228 * 325
= 1235
60
9. 0.43(0.18(4250 )) ≈ $329
10.
144(4.99) 64(4.99 )
−
≈ 11.10
36
36
11. The area of ΔAEC plus the area of ΔEBC is half the area of rectangle ABCD. Furthermore, the area of
1
ΔAEC is twice the area of ΔEBC . Thus, the fraction of the area of the rectangle inside ΔAEC is .
3
12. Let x be the perpendicular distance from point P to each of the three sides. Then
(25
equals the area of ΔABC . The area of ΔABC is
13. Tile the floor as shown in the diagram.
9
2
25 x 25 x 14 x
+
+
2
2
2
− 7 2 )(14 )
21
= 168 and x =
.
4
2
2008 State Math Contest
Wake Technical Community College
Geometry Test
14. Let triangle CDA be the right triangle formed by Cy (point C), the base of the building (point D), and Al
(point A) and triangle CDB be the right triangle formed by Cy, the base of the building, and Bob (point
B). CD = 12 feet and m∠CAD = 30° so CA = 24 feet. Triangle CAB is an isosceles triangle. AB = 24
feet.
15. Al can not say he is an alternator unless he is a knave or an alternator. If Al is a knave then his second
statement is a lie which means that Bo is alternator. If Al is an alternator, then his first statement is true
so his second must be a lie. Once again Bo must be an alternator.
16. Let ABCD be the trapezoid with sides AB = BC = CD = 4 inches and side AD = 8 inches. BC AD and
m∠DAB = m∠ADC = 45° . Hence, the height to side AD is 2 3 inches and the area of trapezoid
ABCD is 12 3 sq inches.
17. Let y be the length of the side of the smaller cube, then ( y + 4) − y 3 = 259 . So, y = 2.5 meters and the
sum of the two lengths is 9 meters.
3
18. The new polyhedron has 12 vertices (the midpoints of each of the twelve edges of a cube) and 24 edges
(four edges in each of the six faces of the cube). Hence, the new polyhedron must have 24 + 2 − 12 = 14
faces.
19. Triangle APQ is a 30-60-90 triangle, AP = 15, and QA = 15 3 so BA = 7.5 3 . Hence,
(
PB = 15 2 + 7.5 3
)
2
≈ 19.84 ft
20. (1.2 )(1.3)(.6 ) = 0.936 . So the volume change is a 6.4% decrease.
21. Let x be the radius of the semicircle, then x 2 − 49 = ( x − 1) . So x = 25 feet.
2
22. Let x be the radius of circle B. The line through the center of A parallel to the x-axis and the line through
the center of B parallel to the y-axis intersect in a right angle at point C. The centers of the two circles
and point C form a 45-45-90 right triangle with sides of length (x − 1) unit and hypotenuse of length
(x + 1) .
So x = 3 + 2 2 units.
23. The circle formed by the new belt has radius equal to the radius of the earth +
100
feet or 15.92 feet.
2π
That is plenty of room for a tractor trailer.
24. 5
((1.35)(1.35 − .85)(1.35 − .85)(1.35 − 1)) ≈ 1.72 sq yards
25. The four quadrants in counterclockwise order beginning with the first quadrant are Colorado, Utah,
Arizona, and New Mexico. Wyoming lies in both the first and the second quadrant.
10
2008 State Math Contest
Wake Technical Community College
Geometry Test
66. Let S be the point in the rectangular lot ABCD where the spring is located. Let DS = x, let h1 be the
perpendicular distance from S to side AB, let h2 be the perpendicular distance from S to side BC, let h3
be the perpendicular distance from S to side CD, and let h4 be the perpendicular distance from S to side
2
2
2
2
DA. Then h1 + h22 = 30 2 , h2 + h32 = 40 2 , h1 + h42 = 50 2 , and h3 + h42 = x 2 . Hence,
40 2 + 50 2 = 30 2 + x 2 so x is approximately 57 feet.
67. The list of twelve possibilities are {1,1,200} , {1,2,100}, {1,4,50}, {1,5,40}, {1,8,25} , {1,10,20}, {2,2,50} ,
{2,4,25} , {2,5,20}, {2,10,10}, {4,5,10}, and {5,5,8}.
68. The area of triangle ABC is 4.5 square units. The area of each of the circular arcs ABE and BCD are
1.125 π . The sum of the area of the two circular arcs is the area of the triangle plus the area of the
unshaded region of the triangle. Hence, 2.25π − 4.5 is the area of the unshaded region giving the area
of the shaded region equal to 9 − 2.25π ≈ 1.93 square units.
2
69. Let d be the original diameter of the can and d 1 be the new diameter. Hence, 0.92d 2 = d1 so
d1 = 0.92d ≈ 0.959d or decreased by 4.1%.
70. The area of the triangle is
length 48 cm is 21 cm.
((56)(56 − 29)(56 − 35)(56 − 48)) = 504 sq cm so the height to the side of
11