Question
A cylinder has a surface area of 54π cm2. If it were dilated using a scale factor of 1/2, what is the surface area of the new cylinder?
Responses
13.5π cm2
13.5π cm2 - not selected, this is the correct answer
6.75π cm2
6.75π cm2 - no response given
108π cm2
108π cm2 - incorrect
27π cm2
Responses
13.5π cm2
13.5π cm2 - not selected, this is the correct answer
6.75π cm2
6.75π cm2 - no response given
108π cm2
108π cm2 - incorrect
27π cm2
Answers
GPT-4o mini
To find the surface area of the new cylinder after it has been dilated using a scale factor of \( \frac{1}{2} \), we can use the property that the surface area of a three-dimensional shape changes with the square of the scale factor.
1. **Original Surface Area**: The original surface area of the cylinder is \( 54\pi \) cm².
2. **Scale Factor**: The scale factor for the dilation is \( \frac{1}{2} \).
3. **Change in Surface Area**: The new surface area after dilation can be calculated using the formula:
\[
\text{New Surface Area} = (\text{Scale Factor})^2 \times \text{Original Surface Area}
\]
Substituting the values:
\[
\text{New Surface Area} = \left(\frac{1}{2}\right)^2 \times 54\pi = \frac{1}{4} \times 54\pi = \frac{54\pi}{4} = 13.5\pi \text{ cm}^2
\]
So, the surface area of the new dilated cylinder is \( 13.5\pi \) cm². The correct answer is indeed:
**13.5π cm²**
1. **Original Surface Area**: The original surface area of the cylinder is \( 54\pi \) cm².
2. **Scale Factor**: The scale factor for the dilation is \( \frac{1}{2} \).
3. **Change in Surface Area**: The new surface area after dilation can be calculated using the formula:
\[
\text{New Surface Area} = (\text{Scale Factor})^2 \times \text{Original Surface Area}
\]
Substituting the values:
\[
\text{New Surface Area} = \left(\frac{1}{2}\right)^2 \times 54\pi = \frac{1}{4} \times 54\pi = \frac{54\pi}{4} = 13.5\pi \text{ cm}^2
\]
So, the surface area of the new dilated cylinder is \( 13.5\pi \) cm². The correct answer is indeed:
**13.5π cm²**