To determine which approximations can be used to represent \(\pi\), we'll analyze each given value and compare it to the actual value of \(\pi\). We aim to see if the difference between each approximation and \(\pi\) is within an acceptable threshold.
The value of \(\pi\) is approximately \(3.14159\).
Here are the given approximations:
1. \(\frac{22}{7}\)
2. \(3.14\)
3. \(4\)
4. \(\frac{355}{113}\)
5. \(3\)
6. \(\frac{333}{106}\)
### Calculations and Comparisons:
1. \(\frac{22}{7}\)
- Calculation: \(\frac{22}{7} \approx 3.142857\)
- Absolute difference from \(\pi\): \(|3.142857 - 3.14159| \approx 0.001267\)
2. \(3.14\)
- Absolute difference from \(\pi\): \(|3.14 - 3.14159| = 0.00159\)
3. \(4\)
- Absolute difference from \(\pi\): \(|4 - 3.14159| \approx 0.85841\)
4. \(\frac{355}{113}\)
- Calculation: \(\frac{355}{113} \approx 3.141593\)
- Absolute difference from \(\pi\): \(|3.141593 - 3.14159| \approx 0.000003\)
5. \(3\)
- Absolute difference from \(\pi\): \(|3 - 3.14159| = 0.14159\)
6. \(\frac{333}{106}\)
- Calculation: \(\frac{333}{106} \approx 3.141509\)
- Absolute difference from \(\pi\): \(|3.141509 - 3.14159| \approx 0.000081\)
### Conclusion:
For an approximation to be considered valid, its absolute difference from \(\pi\) should be reasonably small. Based on the calculated differences, the following approximations are close enough to \(\pi\) to be considered:
- \(\frac{22}{7}\)
- \(3.14\)
- \(\frac{355}{113}\)
- \(\frac{333}{106}\)
The approximations that can be used to represent \(\pi\) are:
- \(\frac{22}{7}\)
- \(3.14\)
- \(\frac{355}{113}\)
- [tex]\(\frac{333}{106}\)[/tex]
