Determine a mathematical model that represents the data by examining the patterns in the table.

# of folds : 0, 1, 2, 3, 4, 5, 6
# of sections: 1, 2, 4, 8, 16, 32, 64
Expanded Expression: 1, 1x2, 2x2, 2x4, 4x4, 8x4, 8x8
Simplified Expression: 2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6

1 answer

The data you've provided relates the number of folds to the number of sections produced, which appears to follow a clear pattern. Let's analyze the information:

  1. Table of Values:

    • # of folds (n): 0, 1, 2, 3, 4, 5, 6
    • # of sections (S): 1, 2, 4, 8, 16, 32, 64
  2. Observing Patterns:

    • The number of sections doubles as the number of folds increases by 1.
    • This indicates that the relationship between the number of folds and the number of sections can be represented by an exponential function.
  3. Expanded and Simplified Expressions:

    • The Expanded Expression provided shows how the number of sections can also be represented by the product of previous sections combined with the number of folds:

      • \( 1 \) (for 0 folds)
      • \( 1 \times 2 \) (for 1 fold)
      • \( 2 \times 2 \) (for 2 folds)
      • \( 2 \times 4 \) (for 3 folds)
      • \( 4 \times 4 \) (for 4 folds)
      • \( 8 \times 4 \) (for 5 folds)
      • \( 8 \times 8 \) (for 6 folds)
    • The Simplified Expression shows the exponential growth:

      • \( 2^0 = 1 \)
      • \( 2^1 = 2 \)
      • \( 2^2 = 4 \)
      • \( 2^3 = 8 \)
      • \( 2^4 = 16 \)
      • \( 2^5 = 32 \)
      • \( 2^6 = 64 \)
  4. Mathematical Model: From the observations above, we can deduce a mathematical model that represents this relationship. The number of sections \( S \) can be expressed as a function of the number of folds \( n \):

\[ S(n) = 2^n \]

Where \( S(n) \) is the number of sections produced after \( n \) folds.

Conclusion: The mathematical model representing the relationship between the number of folds and the number of sections is \( S(n) = 2^n \). This model indicates that for each additional fold, the number of sections produced doubles.