To find the new surface area of a cylinder after dilation using a scale factor of 4, we can follow these steps:
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Original Surface Area Formula: The surface area \( SA \) of a cylinder is given by: \[ SA = 2\pi rh + 2\pi r^2 \] where \( r \) is the radius and \( h \) is the height.
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Effect of Dilation: When an object is dilated by a scale factor \( k \), every linear dimension (including radius and height) is multiplied by \( k \). For a scale factor of 4: \[ r' = 4r \quad \text{and} \quad h' = 4h \]
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New Surface Area: The new surface area \( SA' \) will be calculated using the new radius and height: \[ SA' = 2\pi (r')(h') + 2\pi (r')^2 \] Substituting the values: \[ SA' = 2\pi (4r)(4h) + 2\pi (4r)^2 \]
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Calculating Individual Terms:
- Surface area of the side: \[ 2\pi (4r)(4h) = 2\pi (16rh) = 32\pi rh \]
- Surface area of the top and bottom: \[ 2\pi (4r)^2 = 2\pi (16r^2) = 32\pi r^2 \]
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New Surface Area: \[ SA' = 32\pi rh + 32\pi r^2 = 32\pi (rh + r^2) = 32 \times (2\pi rh + 2\pi r^2) \] Since the original surface area \( SA = 2\pi rh + 2\pi r^2 \), we have: \[ SA' = 32 \times SA \]
Assuming the original surface area \( SA \) is \( x \), then: \[ SA' = 32x \]
- Choosing the Correct Option: Since we don't know the original surface area, we can't compute the exact new value. However, the transformation leads us to conclude that the new surface area is 32 times the original surface area.
Given the response options provided, we find:
- If \( SA = 1\pi \), then \( SA' = 32\pi \), which isn't present in the options.
- To find an appropriate match, we can look for an answer that, when multiplied by 32, yields one of the options provided.
Assuming one of the values is \( 1,472π \): \[ x = \frac{1,472\pi}{32} = 46.0\pi \] This suggests that the original surface area could reasonably be \( 46\pi \), indicating that \( SA' = 5,888π \).
Thus, if the original surface area can match \( 1,772\pi \), then the answer must be:
5,888π cm².
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