Question
A space agency is tracking two comets:
Comet A has a tail length of 2.5 × 10⁶ kilometers.
Comet B has a tail length of 7 × 10⁵ kilometers.
Question:
Using complete sentences, explain how many times longer Comet A’s tail is compared to Comet B’s tail. Show your calculations in scientific notation, and explain why using scientific notation makes it easier to perform this comparison instead of using standard form. Include any reasoning about the exponents and coefficients in your explanation.
Comet A has a tail length of 2.5 × 10⁶ kilometers.
Comet B has a tail length of 7 × 10⁵ kilometers.
Question:
Using complete sentences, explain how many times longer Comet A’s tail is compared to Comet B’s tail. Show your calculations in scientific notation, and explain why using scientific notation makes it easier to perform this comparison instead of using standard form. Include any reasoning about the exponents and coefficients in your explanation.
Answers
GPT-4o mini
To determine how many times longer Comet A's tail is compared to Comet B's tail, we start by setting up the comparison using their tail lengths in scientific notation:
- Comet A's tail length: \(2.5 \times 10^6\) kilometers
- Comet B's tail length: \(7 \times 10^5\) kilometers
Next, we can calculate how many times longer Comet A's tail is by dividing the tail length of Comet A by the tail length of Comet B:
\[
\text{Ratio} = \frac{2.5 \times 10^6}{7 \times 10^5}
\]
To perform the division, we can separate the coefficients and the powers of ten:
\[
\text{Ratio} = \frac{2.5}{7} \times \frac{10^6}{10^5}
\]
First, we calculate the coefficient:
\[
\frac{2.5}{7} \approx 0.3571
\]
Next, we perform the division of the exponents:
\[
\frac{10^6}{10^5} = 10^{6-5} = 10^1
\]
Now, we can combine these results:
\[
\text{Ratio} \approx 0.3571 \times 10^1 = 3.571
\]
This means that Comet A's tail is approximately 3.571 times longer than Comet B's tail.
Using scientific notation simplifies the process of comparison because it allows us to handle the significant figures and the powers of ten separately. By isolating the coefficients and subtracting the exponents, we avoid the complications of dealing with large numbers in standard form, which can often lead to errors in direct arithmetic operations. In this instance, focusing on the exponents \(10^6\) and \(10^5\) clearly shows that Comet A is significantly longer due to the difference in their orders of magnitude (where the exponent increases), while the division of the coefficients remains manageable within the range of simple calculators or mental math. This illustrates the efficiency of scientific notation in performing such comparisons in a straightforward manner.
- Comet A's tail length: \(2.5 \times 10^6\) kilometers
- Comet B's tail length: \(7 \times 10^5\) kilometers
Next, we can calculate how many times longer Comet A's tail is by dividing the tail length of Comet A by the tail length of Comet B:
\[
\text{Ratio} = \frac{2.5 \times 10^6}{7 \times 10^5}
\]
To perform the division, we can separate the coefficients and the powers of ten:
\[
\text{Ratio} = \frac{2.5}{7} \times \frac{10^6}{10^5}
\]
First, we calculate the coefficient:
\[
\frac{2.5}{7} \approx 0.3571
\]
Next, we perform the division of the exponents:
\[
\frac{10^6}{10^5} = 10^{6-5} = 10^1
\]
Now, we can combine these results:
\[
\text{Ratio} \approx 0.3571 \times 10^1 = 3.571
\]
This means that Comet A's tail is approximately 3.571 times longer than Comet B's tail.
Using scientific notation simplifies the process of comparison because it allows us to handle the significant figures and the powers of ten separately. By isolating the coefficients and subtracting the exponents, we avoid the complications of dealing with large numbers in standard form, which can often lead to errors in direct arithmetic operations. In this instance, focusing on the exponents \(10^6\) and \(10^5\) clearly shows that Comet A is significantly longer due to the difference in their orders of magnitude (where the exponent increases), while the division of the coefficients remains manageable within the range of simple calculators or mental math. This illustrates the efficiency of scientific notation in performing such comparisons in a straightforward manner.