Download 2006 State Math Contest Wake Technical Community College

2006 State Math Contest
Wake Technical Community College
Geometry Test
1. The manufacturers of a certain pen claim that it can draw a line 1 km long before it runs dry. If the line
it draws is 0.4 mm wide, then what is the area, in square meters, that the pen is expected to cover?
a. 4000 m²
b. 400 m²
c. 40 m²
d. 4 m²
e. 0.4 m²
2. The interior angles of a convex regular polygon measure 168˚. How many sides does the polygon have?
a. 12
b. 15
c. 20
d. 30
e. 36
3. Andy did a survey of the students in his math class and found that 4 students walk to school, 15 students
ride the bus to school, 6 students drive to school, and 7 students ride to school with their parents. When
he made a pie graph of the data, what was the degree measure of the sector representing the students
who walk to school?
a. 4˚
b. 40˚
c. 90˚
d. 45˚
e. 50˚
4. Let A and B be two cubes such that the ratio of the volume of A to the volume of B is 8. What is the
ratio of the surface area of A to the surface area of B?
a. 16
b. 12
c. 24
d. 14
e. 4
5. A man walks 3 miles east, then 3 miles north, and then 2 miles northwest. How far is he from his
starting point to the nearest tenth of a mile?
a. 4.7 mi
b. 6.3 mi
c. 4.5 mi
d. 4.6 mi
e. 6.1 mi
6. Which of the following is the converse of “If I do all my homework, then I will get a good grade.”?
a. If I get a good grade, then I did all my homework.
b. If I do not do all my homework, then I will not get a good grade.
c. If I do not get a good grade, then I did not do all my homework.
d. If I do not do some of my homework, then I will not get a good grade.
e. If I do not get a good grade, then I did not do some of my homework.
1
2006 State Math Contest
Wake Technical Community College
Geometry Test
7. A large square has a smaller square cut from its corner in such a way that the area of the square removed
equals the area of the remaining region. If x is the length of the side of the removed square and x + y is
x
the length of the original square, then what is ?
y
a. 1
b. 1 + 2
c. 2
d.
2 −1
e.
1
2
8. The following four statements are found on a card and only these four statements:
On this card exactly one statement is false.
On this card exactly two statements are false.
On this card exactly three statements are false.
On this card exactly four statements are false.
Exactly how many of the statements are false?
a. 0
b. 1
c. 2
d. 3
e. 4
9. In the figure at the right, a wooden cube has edges of length 3 meters. Square holes
1 meter on each side, centered in each face, are cut through to the opposite face.
The edges of the holes are parallel to the edges of the cube. What is the entire
surface area including the inside in square meters?
a. 54 m²
b. 72 m²
c. 76 m²
d. 84 m²
e. 86 m²
10. Find the area of a regular hexagon inscribed in a circle of radius 1 inch to the nearest
hundredth.
a. 2.57 in²
b. 2.58 in²
c. 2.59 in²
2
d. 2.60 in²
e. 2.61 in²
2006 State Math Contest
Wake Technical Community College
Geometry Test
11. Let 4, 7, and c be the lengths of the sides of a triangle. If c is an integer, then what is the difference
between the largest and smallest possible value of c?
a. 4
b. 5
c. 6
d. 7
e. 8
12. Let C be a cube where the length in feet of its longest diagonal is the same as its volume in cubic feet.
What is the length in feet of each side to the nearest thousandth?
a. 1.732 ft
b. 1.437 ft
c. 1.416 ft
d. 1.316 ft
e. 1.246 ft
13. What is the maximum number of acute angles that any convex polygon can have?
a. 2
b. 3
c. 4
d. 5
e. no limit
14. The Pythagorean Club of a local high school designed a membership pin made up
of two circles, a square, and an equilateral triangle (see diagram). The square is
inscribed in the first circle, the second circle is inscribed in the square, and the
equilateral triangle is inscribed in the second circle. If the diameter of the larger
circle is three inches, what is the length of a side of the equilateral triangle to
the nearest hundredth of an inch?
a. 1.69 in
b. 1.72 in
c. 1.92 in
d. 1.73 in
e. 1.84 in
15. In the figure, ∠ABC and ∠BDA are both right angles. If AB + BC = 35 and AC + BD = 37 , what is
the value of BD to the nearest tenth?
B
C
a. 10.5
b. 11.2
D
A
c. 12.0
3
d. 12.5
e. 13.2
2006 State Math Contest
Wake Technical Community College
Geometry Test
16. In the figure, ∠EAB and ∠ABC are right angles, AB = 4, BC = 6, AE = 8, and AC and BE intersect at
D. What is the difference between the areas of ΔADE and ΔBDC ?
E
C
D
A
a. 10
b. 8
B
c. 6
d. 4
e. 2
17. The point A ( −3, 2 ) is rotated 90° clockwise around the origin to point B. Point B is then reflected over
the line y = x to point C. What are the coordinates of C?
a.
( 2,3)
b.
( 2, −3)
c.
( −3, −2 )
d.
( 3, 2 )
e. ( 3, −2 )
18. An equilateral triangle and a regular hexagon have the same perimeter. What is the ratio of the area of
the hexagon to the area of the triangle?
a. 1.5
b. 2
c.
3 3
2
d.
4 3
3
e. 1
19. The sum of the interior angles of a convex polygon is less than 2006°. What is the largest possible
number of sides of the polygon?
a. 10
b. 12
c. 13
d. 14
e. 15
20. A pyramid has a square base with sides of length 20 m and height 10 m. The length of the sloping edges
are of equal length. How long is a sloping edge to the nearest hundredth of a meter?
a. 14.14 m
b. 15.37 m
c. 18.62 m
4
d. 20.14 m
e. 17.32 m
2006 State Math Contest
Wake Technical Community College
Geometry Test
R
21. An archway is constructed on a straight base PQ with two circular arcs PR and
QR. Arc PR has center Q and arc QR has center P. If PQ is 2 yards, what is the
area of the archway to the nearest tenth?
P
a. 2.5 yd²
b. 2.4 yd²
c. 1.8 yd²
d. 2.0 yd²
Q
e. 1.9 yd²
22. During a windstorm, a 32-foot vertical flag pole positioned on the ground cracks in such a way that the
top of the pole touches the ground 12 feet from the base of the pole. How many feet above the ground
did the break occur?
a. 12.5 ft
b. 13 ft
c. 13.75 ft
d. 12.75 ft
e. 13.5 ft
23. What is the first time after 4 o’clock that the hands of the clock make an angle of 65˚?
a. 4:06
b. 4:07
c. 4:08
d. 4:09
e. 4:10
24. What is the maximum number of pieces into which a pizza can be cut by making 6 cuts?
a. 18
b. 20
c. 22
d. 21
e. 19
25. In the diagram, the square has two of its vertices on the circle, and the other two lie on a
tangent to the circle. What is the ratio of the area of the square to the area of the circle?
a.
5π
8
b.
64
25π
c.
8
5π
5
d.
5
3π
e.
25
9π
2006 State Math Contest
Wake Technical Community College
Geometry Test
SHORT ANSWER
Place the answer in the appropriate space.
66. If the sides of a right triangle are 6 inches, 8 inches, and 10 inches the area is 24 square inches and its
perimeter is 24 inches. There is one other Pythagorean triple that has this same property (where the
number of square units in the area is equal to the number of units in the perimeter). What is the other
Pythagorean triple?
67. Batman and Robin each order a pizza. The circumference of Batman’s pizza is 20 percent greater than
the circumference of Robin’s pizza. The area of Batman’s pizza is what percentage greater than the area
of Robin’s pizza?
68. A regular polygon with 2006 sides has a perimeter of 1 unit. The area of this regular polygon is closest
1
to which unit fraction: ?
n
69. A box whose inside measurements are 4 ft. by 4 ft. by 4 ft. is packed with cans that are 2 ft. high and
have a diameter of 6 in. When the box is fully packed with cans, how much space is wasted in the box
to the nearest cubic foot?
70. Let AB be a diameter of a circle and C be a point such that BC is tangent to the circle and
m∠BAC = 30° . Let D be the intersection point of the circle and AC . If CD = 3 cm, then what is the
radius of the circle?
A
D
B
C
6
2006 State Math Contest
Wake Technical Community College
Geometry Test
1. e
2. d
3. d
4. e
5. a
6. a
7. b
8. d
9. b
10. d
11. c
12. d
13. b
14. e
15. c
16. d
17. d
18. a
19. c
20. e
21. a
22. c
23. e
24. c
25. b
66. 5-12-13
67. 44%
1
68.
sq units
13
69. 14 ft³
70. 3 cm
7
2006 State Math Contest
Wake Technical Community College
Geometry Test
1. 1000 × 0.0004 = 0.4 m²
2. 180° − 168° = 12° and 360°/12° = 30 sides
3.
4
360° = 45°
32
4. If the ratio of the volumes is 8 then the ratio of the sides is 2. Hence the ratio of the surface areas is 4.
5.
( )
22 + 3 2
2
= 22 ≈ 4.7 miles
6. The converse is “If I get a good grade, then I did all my homework.”
2
⎛x⎞
x2
x
x
x
we get 1 + 2 .
7. x = y + 2 xy so 2 = 1 + 2 or ⎜⎜ ⎟⎟ − 2 − 1 = 0 . Solving for
y
y
y
y
⎝ y⎠
2
2
8. The only true statement on the card is “On this card exactly three statements are false.”
9. On the outside of the cubes the surface area is 6 times 8 m² or 48 m². On the inside the surface area is 6
times 4 m² or 24 m². So the total is 72 m².
12 3
10. 6
≈ 2.60 in²
4
11. The smallest c can be is 4 and the largest c can be is 10. So the difference is 6.
12. Let x be the side of the cube then x 3 = x 3 . Hence x = 4 3 and this is approximately 1.316 ft.
13. A triangle can have 3 acute angles so the answer is at least 3. For any convex polygon with four or more
sides if there were more than 3 acute angles then one of the other angles would need to be more than
180° which contradicts the polygon being convex.
14. The radius of the large circle is 1.5 inches so the radius of the small circle is
3 2
9 2
inches. So the height of the equilateral triangle is
inches and the
4
8
3 6
side is
inches or 1.84 in.
4
15. ( AB + BC ) = 1225 or AB 2 + 2( AB )(BC ) + BC 2 = 1225 . Similarly AC 2 + 2( AC )(BD ) + BD 2 = 1369 .
Since ( AC )(BD ) = ( AB )(BC ) and AB 2 + BC 2 = AC 2 , we get by substitution 1225 + BD 2 = 1369 or BD
is 12.0.
2
8
2006 State Math Contest
Wake Technical Community College
Geometry Test
16. ΔADE and ΔBDC are similar triangles. Let x be the height of ΔADE , then the height of ΔBDC is
3x
3x
16
⎛ 16 ⎞ ⎛ 12 ⎞
. Solving x +
= 4 we get x is
. 4⎜ ⎟ − 3⎜ ⎟ = 4 .
4
4
7
⎝7⎠ ⎝7⎠
17. When A ( −3, 2 ) is rotated 90° clockwise around the origin the new point B is B(2,3) . So when reflected
over line y = x the point C is C (3,2 ) .
18. Let x be the side of the regular hexagon, then the equilateral triangle will have side length 2x. Thus the
6x 2 3
6x 2 3
4
=
= 1.5
areas satisfy the ratio
(2 x )2 3 4 x 2 3
4
19. Let n be the number of sides of a convex polygon, then the sum of the interior angles of the polygon is
180°n − 360° . Solving 180°n − 360° < 2006° we get 13 sides.
20. Let ABCD be the square base of the pyramid and E be the apex. Let F be the center of the square base,
then EF = 10 m and AF = 10 2 . ΔAFE is a right triangle so EF 2 + AF 2 = AE 2 . Solving we get AE
is approximately 17.32 m.
2
⎛ 60 2 ⎞ 2 3
(
21. ΔPQR is an equilateral triangle. Therefore, the area of the archway is 2⎜
2 π )⎟ −
≈ 2.5 yd².
4
⎝ 360
⎠
22. Let x be the height above the ground where the break occurred, then x 2 + 12 2 = (32 − x ) . Solving for x
we get 13.75 feet.
2
23. Every minute the minute hand moves 6° and the hour hand moves and the hour hand moves 0.5°. Let x
be the number of minutes after 4 o’clock that the hands make an angle of 65°, then
65° = 120° − 6° x + 0.5° x . Solving for x we get 10 minutes so the time is 4:10.
24. The maximum number of pieces into which a pizza can be cut by making 6 cuts is the same as the
n(n + 1)
number of regions the plane can be cut into by 6 distinct intersecting lines. The formula is
+1
2
which can be derived by considering the pattern 0 lines is 1 region; 1 line is 2 regions; 2 lines is 4
regions; and 3 lines is 7 regions. The pattern is the nth line adds an additional n regions.
25. Let x be the length of the side of the square and r be the radius of the circle. Let O be the center of the
circle, ABCD be the square with side AB lying fully inside the circle, and M be the midpoint of side AB.
2
x
⎛ x⎞
2
Then ΔOMB is a right triangle with sides , r, and x − r. Solving ⎜ ⎟ + ( x − r ) = r 2 for x we get
2
⎝2⎠
64
8
.
x = r . Thus the required ratio is
25π
5
9
2006 State Math Contest
Wake Technical Community College
66. Since 5 + 12 + 13 = 30 and
Geometry Test
1
(5)(12) = 30 . Then the other Pythagorean triple with the required property
2
is the 5 – 12 – 13 triple.
67. Let x be the radius of Robin’s pizza, then 1.2x is the radius of Batman’s pizza. Hence, the area of
Batman’s pizza is 1.44 times the area of Robin’s pizza or 44% greater.
68. A regular polygon with 2006 sides is very close to a circle. A circle with perimeter or more precisely
1
1
1
circumference 1 unit has a radius of
. The area of this circle is
which is approximately
.
2π
4π
13
69. 4 3 − 128(2 )(0.25) π ≈ 14 ft³
2
70. Since AB is a diameter then ∠BDA and ∠BDC are right angles. Moreover ΔABC , ΔABD , and
ΔBDC are 30°-60°-90° triangles. Hence BC is 2 3 cm, BD is 3 cm, and AB is 6 cm. Therefore the
radius of the circle is 3 cm.
A
D
B
C
10