To find the new surface area of a dilated cylinder, we first need to determine the new dimensions after applying a scale factor of 4.
Original dimensions of the cylinder:
- Height (h) = 15 cm
- Radius (r) = 8 cm
After dilation with a scale factor of 4:
- New height (h') = 4 * 15 cm = 60 cm
- New radius (r') = 4 * 8 cm = 32 cm
Next, we can use the surface area formula for a cylinder, which is:
\[ SA = 2\pi rh + 2\pi r^2 \]
Now we can substitute the new dimensions into this formula.
Calculating the new surface area:
-
Calculate \(2\pi r'h'\): \[ 2\pi (32)(60) = 2\pi (1920) = 3840\pi \]
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Calculate \(2\pi r'^2\): \[ 2\pi (32^2) = 2\pi (1024) = 2048\pi \]
Now, combine both parts to find the total surface area: \[ SA' = 3840\pi + 2048\pi = 5888\pi \]
Therefore, the new surface area of the dilated cylinder is \(5,888\pi , \text{cm}^2\).
The correct response is: 5,888π cm²