Question

the cylinder has 8cm diameter and a height of 15 cm.
If the cylinder were dilated using a scale factor of 4, what would be the new surface area? the formula for the suface area of a cylinder is S A = 2pirh+2pir^2

A. 23pi cm ^2
B. 5,888pi cm^2
C. 1,472pi cm^2
D. 368pi cm^2

Answer 1

To find the new surface area of a dilated cylinder, we can follow these steps:

1. **Calculate the original surface area of the cylinder using the formula:**
\[
SA = 2\pi rh + 2\pi r^2
\]

Where:
- \( r \) is the radius
- \( h \) is the height

The diameter of the cylinder is 8 cm, so the radius \( r \) is:
\[
r = \frac{8}{2} = 4 \text{ cm}
\]

The height \( h \) is 15 cm.

Plug these values into the surface area formula:
\[
SA = 2\pi (4)(15) + 2\pi (4)^2
\]

Calculating each term:
- \( 2\pi (4)(15) = 120\pi \)
- \( 2\pi (4)^2 = 32\pi \)

Now combine the two terms:
\[
SA = 120\pi + 32\pi = 152\pi \text{ cm}^2
\]

2. **Determine the new dimensions after dilation:**
If the cylinder is dilated by a scale factor of 4:
- New radius \( r' = 4 \times 4 = 16 \text{ cm} \)
- New height \( h' = 4 \times 15 = 60 \text{ cm} \)

3. **Calculate the new surface area of the dilated cylinder:**
\[
SA' = 2\pi r'h' + 2\pi (r')^2
\]

Substitute the new values:
\[
SA' = 2\pi (16)(60) + 2\pi (16)^2
\]

Calculating each term:
- \( 2\pi (16)(60) = 1920\pi \)
- \( 2\pi (16)^2 = 512\pi \)

Now combine the two terms:
\[
SA' = 1920\pi + 512\pi = 2432\pi \text{ cm}^2
\]

4. **Relate area scaling to the scale factor:**
When a three-dimensional shape is scaled by a factor \( k \), the surface area is scaled by \( k^2 \). In this case, the scale factor is \( 4 \), so we have:
\[
\text{New Surface Area} = k^2 \times \text{Original Surface Area} = 16 \times 152\pi = 2432\pi \text{ cm}^2
\]

Thus, the new surface area of the cylinder after dilation is:
\[
\boxed{2432\pi \text{ cm}^2}
\]

Since \( 2432\pi \text{ cm}^2 \) is not one of the provided answer choices, please double-check your choices. If the options correspond to a different problem or calculation method, those results could differ.