Question

If the cylinder were dilated using a scale factor of 4, what would be the new surface area? The formula for the surface area of a cylinder is SA=2πrh+2πr2 . (1 point) Responses 1,472π cm2 1 comma 472 pi cm squared 5,888π cm2 5,888 pi cm squared 23π cm2 23 pi cm squared 368π cm2

Answer 1

To find the new surface area of a cylinder after dilation using a scale factor of 4, we can follow these steps:

1. **Original Surface Area Formula**: The surface area \( SA \) of a cylinder is given by:
\[
SA = 2\pi rh + 2\pi r^2
\]
where \( r \) is the radius and \( h \) is the height.

2. **Effect of Dilation**: When an object is dilated by a scale factor \( k \), every linear dimension (including radius and height) is multiplied by \( k \). For a scale factor of 4:
\[
r' = 4r \quad \text{and} \quad h' = 4h
\]

3. **New Surface Area**: The new surface area \( SA' \) will be calculated using the new radius and height:
\[
SA' = 2\pi (r')(h') + 2\pi (r')^2
\]
Substituting the values:
\[
SA' = 2\pi (4r)(4h) + 2\pi (4r)^2
\]

4. **Calculating Individual Terms**:
- Surface area of the side:
\[
2\pi (4r)(4h) = 2\pi (16rh) = 32\pi rh
\]
- Surface area of the top and bottom:
\[
2\pi (4r)^2 = 2\pi (16r^2) = 32\pi r^2
\]

5. **New Surface Area**:
\[
SA' = 32\pi rh + 32\pi r^2 = 32\pi (rh + r^2) = 32 \times (2\pi rh + 2\pi r^2)
\]
Since the original surface area \( SA = 2\pi rh + 2\pi r^2 \), we have:
\[
SA' = 32 \times SA
\]

Assuming the original surface area \( SA \) is \( x \), then:
\[
SA' = 32x
\]

6. **Choosing the Correct Option**: Since we don't know the original surface area, we can't compute the exact new value. However, the transformation leads us to conclude that the new surface area is **32 times the original** surface area.

Given the response options provided, we find:
- If \( SA = 1\pi \), then \( SA' = 32\pi \), which isn't present in the options.
- To find an appropriate match, we can look for an answer that, when multiplied by 32, yields one of the options provided.

Assuming one of the values is \( 1,472π \):
\[
x = \frac{1,472\pi}{32} = 46.0\pi
\]
This suggests that the original surface area could reasonably be \( 46\pi \), indicating that \( SA' = 5,888π \).

Thus, if the original surface area can match \( 1,772\pi \), then the answer must be:

**5,888π cm²**.