To find the new surface area of the cylinder after dilation, we first need to calculate the original surface area using the given dimensions and then apply the scale factor.
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Original dimensions:
- Height \( h = 15 \) cm
- Radius \( r = 8 \) cm
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Surface area formula: \[ \text{SA} = 2\pi rh + 2\pi r^2 \]
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Calculate original surface area:
- Calculate \( 2\pi rh \): \[ 2\pi (8)(15) = 240\pi \text{ cm}^2 \]
- Calculate \( 2\pi r^2 \): \[ 2\pi (8^2) = 2\pi (64) = 128\pi \text{ cm}^2 \]
- Total surface area: \[ \text{SA} = 240\pi + 128\pi = 368\pi \text{ cm}^2 \]
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Dilate the dimensions: When a shape is dilated by a scale factor, all dimensions are multiplied by that scale factor.
- New radius \( r' = 4 \times 8 = 32 \) cm
- New height \( h' = 4 \times 15 = 60 \) cm
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Calculate the new surface area: \[ \text{SA'} = 2\pi r'h' + 2\pi (r')^2 \]
- Calculate \( 2\pi r'h' \): \[ 2\pi (32)(60) = 3840\pi \text{ cm}^2 \]
- Calculate \( 2\pi (r')^2 \): \[ 2\pi (32^2) = 2\pi (1024) = 2048\pi \text{ cm}^2 \]
- Total new surface area: \[ \text{SA'} = 3840\pi + 2048\pi = 5888\pi \text{ cm}^2 \]
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Final answer: The new surface area after dilation is \( 5,888\pi \text{ cm}^2 \).
So the correct response is:
5,888π cm²