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. One paragraph
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. One paragraph
Answers
GPT-4o mini
To determine how many times longer Comet A’s tail is compared to Comet B’s tail, we first express the tail lengths in scientific notation: Comet A’s tail length is 2.5 × 10⁶ kilometers, and Comet B’s tail length is 7 × 10⁵ kilometers. To find the ratio of the lengths, we divide Comet A’s tail length by Comet B’s tail length:
\[
\frac{2.5 × 10⁶}{7 × 10⁵} = \frac{2.5}{7} × \frac{10⁶}{10⁵}.
\]
Calculating the coefficients, \( \frac{2.5}{7} \approx 0.3571 \), and for the powers of ten, we subtract the exponents: \( 10^{6-5} = 10^{1} \). Thus, the ratio becomes:
\[
0.3571 × 10^{1} = 3.571 × 10^{0} = 3.571.
\]
This means Comet A’s tail is approximately 3.571 times longer than Comet B’s tail. Using scientific notation simplifies the calculations, particularly when dealing with large numbers, as it allows us to easily separate the coefficients from the powers of ten. By focusing on the exponents, we can quickly determine how the lengths compare geometrically, which is more complicated when using standard numerical forms that could involve larger integers.
\[
\frac{2.5 × 10⁶}{7 × 10⁵} = \frac{2.5}{7} × \frac{10⁶}{10⁵}.
\]
Calculating the coefficients, \( \frac{2.5}{7} \approx 0.3571 \), and for the powers of ten, we subtract the exponents: \( 10^{6-5} = 10^{1} \). Thus, the ratio becomes:
\[
0.3571 × 10^{1} = 3.571 × 10^{0} = 3.571.
\]
This means Comet A’s tail is approximately 3.571 times longer than Comet B’s tail. Using scientific notation simplifies the calculations, particularly when dealing with large numbers, as it allows us to easily separate the coefficients from the powers of ten. By focusing on the exponents, we can quickly determine how the lengths compare geometrically, which is more complicated when using standard numerical forms that could involve larger integers.