Question
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
# 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
Answers
GPT-4o mini
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.
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.