Answer :
Certainly! Let's determine for which of the given ages the approximate age of Earth, [tex]\(5 \times 10^9\)[/tex] years old, could be considered an approximation.
### Given Ages:
1. [tex]\(4,582,100,000\)[/tex] years
2. [tex]\(45,000,000,000\)[/tex] years
3. [tex]\(4.38 \times 10^9\)[/tex] years
4. [tex]\(4.819999999 \times 10^9\)[/tex] years
5. [tex]\(4,923,000,000\)[/tex] years
### Approximate Age:
[tex]\(5 \times 10^9\)[/tex] years
Let's compare each given age to [tex]\(5 \times 10^9\)[/tex] years to determine if they could be reasonable approximations.
### Steps:
1. Convert the Ages:
- Convert each age to a common format (scientific notation) for easier comparison where necessary.
2. Determine the Range:
- An approximation to [tex]\(5 \times 10^9\)[/tex] years would reasonably be within [tex]\( \pm 0.5 \times 10^9 \)[/tex] years. This sets an acceptable range from [tex]\(4.5 \times 10^9\)[/tex] years to [tex]\(5.5 \times 10^9\)[/tex] years.
### Comparison:
Let's compare each age to see if they fall within this range.
1. [tex]\(4,582,100,000\)[/tex] years:
- Scientific notation: [tex]\(4.5821 \times 10^9\)[/tex]
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
2. [tex]\(45,000,000,000\)[/tex] years:
- Scientific notation: [tex]\(4.5 \times 10^{10}\)[/tex]
- This is much greater than [tex]\(5.5 \times 10^9\)[/tex] and way out of our acceptable range.
- Not a valid approximation
3. [tex]\(4.38 \times 10^9\)[/tex] years:
- This is less than [tex]\(4.5 \times 10^9\)[/tex] and falls slightly below our acceptable range.
- Not a valid approximation
4. [tex]\(4.819999999 \times 10^9\)[/tex] years:
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
5. [tex]\(4,923,000,000\)[/tex] years:
- Scientific notation: [tex]\(4.923 \times 10^9\)[/tex]
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
### Conclusion:
Thus, the ages that could be approximations of [tex]\(5 \times 10^9\)[/tex] years are:
- [tex]\(4,582,100,000\)[/tex] years
- [tex]\(4.819999999 \times 10^9\)[/tex] years
- [tex]\(4,923,000,000\)[/tex] years
Which means the valid choices are:
- [tex]\(4,582,100,000\)[/tex] years (Choice A)
- [tex]\(4.819999999 \times 10^9\)[/tex] years (Choice D)
- [tex]\(4,923,000,000\)[/tex] years (Choice E)
So, the correct answers are A, D, and E.
### Given Ages:
1. [tex]\(4,582,100,000\)[/tex] years
2. [tex]\(45,000,000,000\)[/tex] years
3. [tex]\(4.38 \times 10^9\)[/tex] years
4. [tex]\(4.819999999 \times 10^9\)[/tex] years
5. [tex]\(4,923,000,000\)[/tex] years
### Approximate Age:
[tex]\(5 \times 10^9\)[/tex] years
Let's compare each given age to [tex]\(5 \times 10^9\)[/tex] years to determine if they could be reasonable approximations.
### Steps:
1. Convert the Ages:
- Convert each age to a common format (scientific notation) for easier comparison where necessary.
2. Determine the Range:
- An approximation to [tex]\(5 \times 10^9\)[/tex] years would reasonably be within [tex]\( \pm 0.5 \times 10^9 \)[/tex] years. This sets an acceptable range from [tex]\(4.5 \times 10^9\)[/tex] years to [tex]\(5.5 \times 10^9\)[/tex] years.
### Comparison:
Let's compare each age to see if they fall within this range.
1. [tex]\(4,582,100,000\)[/tex] years:
- Scientific notation: [tex]\(4.5821 \times 10^9\)[/tex]
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
2. [tex]\(45,000,000,000\)[/tex] years:
- Scientific notation: [tex]\(4.5 \times 10^{10}\)[/tex]
- This is much greater than [tex]\(5.5 \times 10^9\)[/tex] and way out of our acceptable range.
- Not a valid approximation
3. [tex]\(4.38 \times 10^9\)[/tex] years:
- This is less than [tex]\(4.5 \times 10^9\)[/tex] and falls slightly below our acceptable range.
- Not a valid approximation
4. [tex]\(4.819999999 \times 10^9\)[/tex] years:
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
5. [tex]\(4,923,000,000\)[/tex] years:
- Scientific notation: [tex]\(4.923 \times 10^9\)[/tex]
- This is within the range [tex]\(4.5 \times 10^9\)[/tex] to [tex]\(5.5 \times 10^9\)[/tex].
- Valid approximation
### Conclusion:
Thus, the ages that could be approximations of [tex]\(5 \times 10^9\)[/tex] years are:
- [tex]\(4,582,100,000\)[/tex] years
- [tex]\(4.819999999 \times 10^9\)[/tex] years
- [tex]\(4,923,000,000\)[/tex] years
Which means the valid choices are:
- [tex]\(4,582,100,000\)[/tex] years (Choice A)
- [tex]\(4.819999999 \times 10^9\)[/tex] years (Choice D)
- [tex]\(4,923,000,000\)[/tex] years (Choice E)
So, the correct answers are A, D, and E.
