Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Rectangles
Moe uses a mower to cut his rectangular 90-foot by150-foot lawn. The swath he cuts is 28 inches wide,but he overlaps each cut by 4 inches to make surethat no grass is missed. He walks at the rate of 5000feet per hour while pushing the mower. Which of thefollowing is closest to the number of hours it will takeMoe to mow his lawn?
(A)0.75(B)0.8(C)1.35(D)1.5(E)3
2003 AMC 10 B, Problem #5—2003 AMC 12 B, Problem #4—“Moe cuts about two square feet for each foot hepushes the mower forward.”
Difficulty:
Medium-hard
NCTM Standard:
Geometry Standard for Grades 9–12: Analyze characteristics and properties of
two- and three-dimensional geometric shapes and develop mathematical arguments about geometric
relationships.
Mathworld.com Classification:
Geometry
>
Plane Geometry
>
Miscellaneous Plane Geometry
>
Area
Many
television
screens
are
rectangles
that
are
measured by the length of their diagonals. The ratio
of the horizontal length to the height in a standard
television screen is
4 : 3
.
The horizontal length of
a “27-inch” television screen is closest, in inches, to
which of the following?
Diagonal
Height
(A)
20
(B)
20
.
5
(C)
21
(D)
21
.
5
(E)
22
2003 AMC 10 B, Problem #6
—
2003 AMC 12 B, Problem #5
—
“The height, length, and diagonal are in the ratio
3 :
4 : 5
.”
Solution
Answer (D):
The height, length, and diagonal are in the ratio
3 : 4 : 5
.
The length of the diagonal is 27, so the horizontal length is
4
5
(27) = 21
.
6
inches
.
Difficulty:
Medium-hard
NCTM
Standard:
Geometry Standard for Grades 9–12:
Use trigonometric relationships to
determine lengths and angle measures.
Mathworld.com Classification:
Geometry
>
Trigonometry
>
General Trigonometry
The symbolism
b
x
c
denotes the largest integer not
exceeding
x
. For example,
b
3
c
= 3
, and
b
9
/
2
c
= 4
.
Compute
b
√
1
c
+
b
√
2
c
+
b
√
3
c
+
· · ·
+
b
√
16
c
.
(A)
35
(B)
38
(C)
40
(D)
42
(E)
136
2003 AMC 10 B, Problem #7
—
“Let
(
a
+1)
2
≤
x < a
2
, for some integer
x
and
a
.
b
√
x
c
=
a
-
1
.”
Solution
Answer (B):
The first three values in the sum are 1, the next five are
2, the next seven are 3, and the final one is 4 for a total of
3
·
1 + 5
·
2 + 7
·
3 + 1
·
4 = 38
.
Difficulty:
Medium
NCTM Standard:
Number and Operations Standard for Grades 9–12: Understand meanings of
operations and how they relate to one another.
Mathworld.com Classification:
Number Theory
>
Rounding
>
Floor Function
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