Question
A cylinder has a surface area of 42 π cm2 . If it were dilated using a scale factor of 12 , what is the surface area of the new cylinder?(1 point) Responses 10.5π cm2 10.5π cm2 84π cm2 84π cm2 13.5π cm2 13.5π cm2 21π cm2
Answers
GPT-4o mini
To find the surface area of the new cylinder after dilation, we first need to know that when a three-dimensional object is dilated by a scale factor \( k \), the surface area increases by a factor of \( k^2 \).
In this case, the original surface area of the cylinder is \( 42 \pi \, \text{cm}^2 \), and the scale factor is \( 12 \).
1. Calculate the factor for the surface area increase:
\[
k^2 = 12^2 = 144
\]
2. Calculate the new surface area:
\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 42 \pi \times 144
\]
Now let's do the multiplication:
\[
42 \times 144 = 6048
\]
Thus:
\[
\text{New Surface Area} = 6048 \pi \, \text{cm}^2
\]
However, it appears I misunderstood your request regarding the options. Since none of these match directly with \( 6048 \pi \), I'm going to summarize:
- The correct answer for the new surface area, after multiplying the original by \( 144 \), is \( 6048 \pi \, \text{cm}^2 \).
- If you are looking for an answer among the options provided, it seems there might be an error in the question or options given, as the computed surface area does not align.
If you have only the options provided and need to select one, refer back to your context, as they don’t match the computation for a scale factor of \( 12 \).
In this case, the original surface area of the cylinder is \( 42 \pi \, \text{cm}^2 \), and the scale factor is \( 12 \).
1. Calculate the factor for the surface area increase:
\[
k^2 = 12^2 = 144
\]
2. Calculate the new surface area:
\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 42 \pi \times 144
\]
Now let's do the multiplication:
\[
42 \times 144 = 6048
\]
Thus:
\[
\text{New Surface Area} = 6048 \pi \, \text{cm}^2
\]
However, it appears I misunderstood your request regarding the options. Since none of these match directly with \( 6048 \pi \), I'm going to summarize:
- The correct answer for the new surface area, after multiplying the original by \( 144 \), is \( 6048 \pi \, \text{cm}^2 \).
- If you are looking for an answer among the options provided, it seems there might be an error in the question or options given, as the computed surface area does not align.
If you have only the options provided and need to select one, refer back to your context, as they don’t match the computation for a scale factor of \( 12 \).