Question
8 cm
15 cm
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 + 2πr^2.
a. 23π cm^2
b. 1,472π cm^2
c. 368π cm^2
d. 5,888π cm^2
15 cm
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 + 2πr^2.
a. 23π cm^2
b. 1,472π cm^2
c. 368π cm^2
d. 5,888π cm^2
Answers
Answered by
GPT-4o mini
To find the new surface area of the cylinder after dilating it by a scale factor of 4, we first need to calculate the original surface area using the given dimensions.
The formula for the surface area (SA) of a cylinder is:
\[
SA = 2\pi rh + 2\pi r^2
\]
Given:
- radius \( r = 8 \, \text{cm} \)
- height \( h = 15 \, \text{cm} \)
**Step 1: Calculate the original surface area.**
First, we substitute the values into the formula:
1. Calculate the lateral surface area \( 2\pi rh \):
\[
2\pi rh = 2\pi (8)(15) = 240\pi \, \text{cm}^2
\]
2. Calculate the area of the top and bottom \( 2\pi r^2 \):
\[
2\pi r^2 = 2\pi (8^2) = 2\pi (64) = 128\pi \, \text{cm}^2
\]
3. Combine them to find the total surface area:
\[
SA = 240\pi + 128\pi = 368\pi \, \text{cm}^2
\]
**Step 2: Determine the new dimensions after dilation.**
When the cylinder is dilated using a scale factor of 4, the new dimensions will be:
- New radius \( r' = 4 \times r = 4 \times 8 \, \text{cm} = 32 \, \text{cm} \)
- New height \( h' = 4 \times h = 4 \times 15 \, \text{cm} = 60 \, \text{cm} \)
**Step 3: Calculate the new surface area.**
Now we use the new dimensions into the surface area formula:
1. Calculate the new lateral surface area:
\[
2\pi r'h' = 2\pi (32)(60) = 3840\pi \, \text{cm}^2
\]
2. Calculate the area of the top and bottom with the new radius:
\[
2\pi (r')^2 = 2\pi (32^2) = 2\pi (1024) = 2048\pi \, \text{cm}^2
\]
3. Combine to find the total surface area:
\[
SA' = 3840\pi + 2048\pi = 5888\pi \, \text{cm}^2
\]
Thus, the new surface area of the cylinder after dilation is:
\[
\boxed{5,888\pi \, \text{cm}^2}
\]
Therefore, the answer is (d) \( 5,888\pi \, \text{cm}^2 \).
The formula for the surface area (SA) of a cylinder is:
\[
SA = 2\pi rh + 2\pi r^2
\]
Given:
- radius \( r = 8 \, \text{cm} \)
- height \( h = 15 \, \text{cm} \)
**Step 1: Calculate the original surface area.**
First, we substitute the values into the formula:
1. Calculate the lateral surface area \( 2\pi rh \):
\[
2\pi rh = 2\pi (8)(15) = 240\pi \, \text{cm}^2
\]
2. Calculate the area of the top and bottom \( 2\pi r^2 \):
\[
2\pi r^2 = 2\pi (8^2) = 2\pi (64) = 128\pi \, \text{cm}^2
\]
3. Combine them to find the total surface area:
\[
SA = 240\pi + 128\pi = 368\pi \, \text{cm}^2
\]
**Step 2: Determine the new dimensions after dilation.**
When the cylinder is dilated using a scale factor of 4, the new dimensions will be:
- New radius \( r' = 4 \times r = 4 \times 8 \, \text{cm} = 32 \, \text{cm} \)
- New height \( h' = 4 \times h = 4 \times 15 \, \text{cm} = 60 \, \text{cm} \)
**Step 3: Calculate the new surface area.**
Now we use the new dimensions into the surface area formula:
1. Calculate the new lateral surface area:
\[
2\pi r'h' = 2\pi (32)(60) = 3840\pi \, \text{cm}^2
\]
2. Calculate the area of the top and bottom with the new radius:
\[
2\pi (r')^2 = 2\pi (32^2) = 2\pi (1024) = 2048\pi \, \text{cm}^2
\]
3. Combine to find the total surface area:
\[
SA' = 3840\pi + 2048\pi = 5888\pi \, \text{cm}^2
\]
Thus, the new surface area of the cylinder after dilation is:
\[
\boxed{5,888\pi \, \text{cm}^2}
\]
Therefore, the answer is (d) \( 5,888\pi \, \text{cm}^2 \).