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:
- Calculate the lateral surface area \( 2\pi rh \):
\[ 2\pi rh = 2\pi (8)(15) = 240\pi , \text{cm}^2 \]
- Calculate the area of the top and bottom \( 2\pi r^2 \):
\[ 2\pi r^2 = 2\pi (8^2) = 2\pi (64) = 128\pi , \text{cm}^2 \]
- 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:
- Calculate the new lateral surface area:
\[ 2\pi r'h' = 2\pi (32)(60) = 3840\pi , \text{cm}^2 \]
- Calculate the area of the top and bottom with the new radius:
\[ 2\pi (r')^2 = 2\pi (32^2) = 2\pi (1024) = 2048\pi , \text{cm}^2 \]
- 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) \( 5,888\pi , \text{cm}^2 \).