Question
An astronomer estimated the distance from Earth to Mercury to be 5.9 × 107 miles, and the distance from Earth to Neptune to be 2.7 × 109 miles. He wanted to know how many times farther it is from Earth to Neptune than from Earth to Mercury. Consider his work.
1.
(2.7 × 109)
(5.9 × 107)
2. (
3.0
6.0
) (
109
107
)
3. 0.5 × 102
The astronomer concluded that the distance in miles from Earth to Neptune is 50 times the distance from Earth to Mercury.
Is he correct? If not, what was his mistake?
Yes, he is correct.
No, the numerator and denominator need to be switched in the expression.
No, he needed to round down in his estimation of the coefficients.
No, he needed to multiple the coefficients.
1.
(2.7 × 109)
(5.9 × 107)
2. (
3.0
6.0
) (
109
107
)
3. 0.5 × 102
The astronomer concluded that the distance in miles from Earth to Neptune is 50 times the distance from Earth to Mercury.
Is he correct? If not, what was his mistake?
Yes, he is correct.
No, the numerator and denominator need to be switched in the expression.
No, he needed to round down in his estimation of the coefficients.
No, he needed to multiple the coefficients.
Answers
GPT-4o mini
To determine if the astronomer's conclusion is correct, let's go through the calculations step by step to find how many times farther it is from Earth to Neptune than from Earth to Mercury.
1. Identify the distances:
- Distance from Earth to Mercury: \(5.9 \times 10^7\) miles
- Distance from Earth to Neptune: \(2.7 \times 10^9\) miles
2. To find how many times farther it is from Earth to Neptune than from Earth to Mercury, we set up the expression:
\[
\frac{2.7 \times 10^9}{5.9 \times 10^7}
\]
3. This can be simplified by dividing the coefficients and subtracting the exponents of 10:
\[
\frac{2.7}{5.9} \times 10^{9 - 7} = \frac{2.7}{5.9} \times 10^2
\]
4. Calculate the coefficient:
\[
\frac{2.7}{5.9} \approx 0.4576271 \quad (\text{approximately})
\]
5. Now multiply this by \(10^2 = 100\):
\[
0.4576271 \times 100 \approx 45.76271
\]
6. Rounding this number, we find it is approximately \(46\).
The astronomer stated that the distance from Earth to Neptune is 50 times the distance from Earth to Mercury. However, our calculation shows it is actually approximately 46 times farther.
Thus, the correct answer is:
**No, he needed to multiply the coefficients.**
He initially calculated incorrectly without properly multiplying the coefficients after dividing the distances.
1. Identify the distances:
- Distance from Earth to Mercury: \(5.9 \times 10^7\) miles
- Distance from Earth to Neptune: \(2.7 \times 10^9\) miles
2. To find how many times farther it is from Earth to Neptune than from Earth to Mercury, we set up the expression:
\[
\frac{2.7 \times 10^9}{5.9 \times 10^7}
\]
3. This can be simplified by dividing the coefficients and subtracting the exponents of 10:
\[
\frac{2.7}{5.9} \times 10^{9 - 7} = \frac{2.7}{5.9} \times 10^2
\]
4. Calculate the coefficient:
\[
\frac{2.7}{5.9} \approx 0.4576271 \quad (\text{approximately})
\]
5. Now multiply this by \(10^2 = 100\):
\[
0.4576271 \times 100 \approx 45.76271
\]
6. Rounding this number, we find it is approximately \(46\).
The astronomer stated that the distance from Earth to Neptune is 50 times the distance from Earth to Mercury. However, our calculation shows it is actually approximately 46 times farther.
Thus, the correct answer is:
**No, he needed to multiply the coefficients.**
He initially calculated incorrectly without properly multiplying the coefficients after dividing the distances.