PT129.S4.P4.Q25

PrepTest 129 - Section 4 - Passage 4 - Question 25

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P1

Fractal geometry is a mathematical theory devoted to the study of complex shapes called fractals. ████████ ██ █████ ██████████ ██ ████████ ███ ███ ████ ████████████ ████████ ████████ ███████ ███ ████████ ██ ████████████████ ███ ███████████ ██ █████████ ███████ ██ ████████ ██ █████████████ ███████ ██████ ██ ████ ████ █████ ████ ██████████ █████ █████████ ████ ███ ██████ ██ █ ██████ ███

Spotlight · Fractal geometry
No exact definition of fractals, but they show self-similarity: small details in shape look like the shape as a whole.
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Example · Koch curve
The Koch curve is an example of a fractal. (Details of what the Koch curve looks like. No need to memorize this. Will come back if asked about it.)
P2

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More on the Koch curve · Can be created by computer
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Interesting point about fractal geometry · Simple processes can create complex patterns
This is one of the major reasons people are into fractal geometry.
P3

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Enthusiasts · Believe fractal geometry will rival importance of calculus
Also believe it will allow mathematicians to effectively describe natural forms (such as a cloud).
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Skeptics · Fractal geometry will have lasting ro in math only if it can support theorems and proofs
Passage Style
Critique or debate
Spotlight
Show answer
25.

Each of the following statements █████ ███ ████ █████ ███ ██ ████████ ███████ ████ ███ ███████████ █████ ██ ███ ███████ ███████

a

The total number ██ ███████████ ██ ███ ████ █████ ██ ███ █████ ██ ███ ████████████ ███████ ██ ███ ██████ ██ ███ ███████ ████ ██████ ███ ███ █████████████

Not supported. At each stage, we split a segment into thirds, the middle third is removed and replaced with two line segments positioned to form a protrusion. Since each stage simply involves splitting a segment into thirds, the initial starting length doesn’t change the number of protrusions. A 6 foot line would be split into thirds at the next stage and there would be 1 point formed out of the middle third. A 600 foot line would be split into thirds at the next stage and there would be 1 point formed out of the middle third.

52%
b

The line segments ██ ████ ██████████ █████ ██ ███ ████████████ ██ ███ ████ █████ ███ ███████ ████ ███ ████████ ██ ███ ████████ ██████

Supported, because at each stage a segment is split into thirds; the middle third is removed and replaced with two line segments as long as the removed piece.

9%
c

Theoretically, as the ████ █████ ██ ███████████ ███ ████ ████████ ██████ ██████████ ██████

Supported, because at each stage a segment is split into thirds; the middle third is removed and replaced with two line segments as long as the removed piece. At each stage, the segments are smaller than they were at the previous stage.

4%
d

At every stage ██ ████████████ ███ ████ ██████ ███ ███ ████ ████████ █████████ ██ ███ ██ █████ ███████

Supported. At each stage, the segments composing the curve are equal in length to each other segment at that stage. (Note, (D) is not asserting that the segments are of equal length ACROSS DIFFERENT STAGES. If it were saying that, then (D) would not be a must be true. It’s saying the segments are of equal length WITHIN EACH STAGE.)

31%
e

The length of ███ ████ ████████ ██ ███ ████ █████ ██ ███ █████ ██ ███ ████████████ ███████ ██ ███ ██████ ██ ███ ███████ ████ ██████ ███ ███ █████████████

Supported. If we start with a 3 foot line, then at the next stage, we’d get 1 foot segments. But if we start with a 6 foot line, then at the next stage we’d get 2 foot segments.

4%

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