Isabella’s house has 3 bedrooms. Each bedroom is 12feet long, 10 feet wide, and 8 feet high. Isabella mustpaint the walls of all the bedrooms.Doorways andwindows, which will not be painted, occupy 60 squarefeet in each bedroom. How many square feet of wallsmust be painted?
(A)678(B)768(C)786(D)867(E)876
2007 AMC 10 B, Problem #1—“Paintonlythewalls,thatmeansnoceilingsandfloors.”
Difficulty:
Medium-easy
NCTM Standard:
Geometry Standard: analyze characteristics and properties of two- and three-
dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Mathworld.com Classification:
Geometry
>
Solid Geometry
>
Polyhedra
>
Cubes
Define the operation
?
by
a ? b
= (
a
+
b
)
b
. What is
(3
?
5)
-
(5
?
3)
?
(A)
-
16
(B)
-
8
(C)
0
(D)
8
(E)
16
2007 AMC 10 B, Problem #2
—
“Plug in 3, 5 for a, b.”
Solution
Answer (E):
Since
3
?
5 = (3 + 5)5 = 8
·
5 = 40
and
5
?
3 = (5 + 3)3 =
8
·
3 = 24
, we have
3
?
5
-
5
?
3 = 40
-
24 = 16
.
Difficulty:
Easy
NCTM Standard:
Number and Operations Standard:
Understand meanings of operations and
how they relate to one another.
Mathworld.com Classification:
Recreational Mathematics
>
Cryptograms
>
Cryptarithmetic
A college student drove his compact car 120 mileshome for the weekend and averaged 30 miles pergallon.On the return trip the student drove hisparents’ SUV and averaged only 20 miles per gallon.What was the average gas mileage, in miles per gallon,for the round trip?
2007 AMC 10 B, Problem #3—“Solve for total distance and total number of gallonsused.”
Difficulty:
Medium
NCTM Standard:
Algebra Standard: Analyze change in various contexts.
Mathworld.com Classification:
Geometry
>
Distance
The pointOis the center of the circle circumscribedabout4ABC, with∠BOC= 120◦and∠AOB=140◦, as shown.What is the degree measure of∠ABC?
120
140
A
C
O
B
(A)
35
(B)
40
(C)
45
(D)
50
(E)
60
2007 AMC 10 B, Problem #4
—
“
OA
=
OB
=
OC
.”
Solution
Answer (D):
Since
OA
=
OB
=
OC
, triangles
AOB
,
BOC
, and
COA
are all isosceles. Hence
∠
ABC
=
∠
ABO
+
∠
OBC
=
180
◦
-
140
◦
2
+
180
◦
-
120
◦
2
= 50
◦
.
OR
Since
∠
AOC
= 360
◦
-
140
◦
-
120
◦
= 100
◦
,
the Central Angle Theorem implies that
∠
ABC
=
1
2
∠
AOC
= 50
◦
.
Difficulty:
Medium
NCTM Standard:
Geometry Standard: Analyze characteristics and properties of two- and three-
dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Circles
Geometry
>
Plane Geometry
>
Triangles
>
Special Triangles
>
Other Triangles
>
Triangle
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