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Q
uantitative
R
easoning &
P
roblem
S
olving
78
© 2014 Pacific Crest
M
odel(s)
Exemplars and representations
M
odel
1: T
ruth
T
able
for
a
F
ormula
:
~P ˄ (P → Q)
P Q
~P P → Q ~P ˄ (P → Q)
T T F
T
F
T F F
F
F
F T T
T
T
F F T
T
T
M
odel
2: D
erivation of
the
F
ormula
for
the
V
olume of
a
T
orus
A
torus
is a 3-dimensional object created by rotating a circle around a
point.
The second theorem of Pappus states that the volume of a solid created
by rotating a plane figure about an external axis is equal to the product
of the area of figure and the distance traveled by the center of the figure.
(Reference available online.)
Step
Model
1.
1
2
r
r
Assume is the inner radius of the torus
and is the outer radius of the torus.
Premises
2.
2 1
(
)
r r
−
Then
is the diameter of the circle being rotated.
By definition of diameter
3.
2
2 1
(
)
2
r r
π
−
The area of the circle being rotated is
.
Formula for area of a circle
4.
2 1
(
)
2
r r
+
The center of the rotating circle is at
.
Definition of center of circle
5.
2 1
2 1
2 (
)
(
)
2
r r
r r
π
π
+
+
The distance traveled by the circle is :
.
or
Formula for circumference
6.
2
2 2 1
2 1
(
)
(
)
4
r r
r r
π
+
−
Therefore, the volume of the torus is :
.
Formula for Volume of a three
dimensional object
M
odel
3: L
ogic
P
roblem
: A
pples
and
P
ears
Tom has three boxes of fruit in his barn: one box containing apples, one box containing pears, and
one box containing both apples and pears. Each of the boxes has a label that should describe the
contents of the box, but none of the labels is on the correct box.