To find the new surface area of the cylinder after dilation with a scale factor of 4, we first need to understand how dilation affects the dimensions of the cylinder.
-
Original dimensions of the cylinder:
- Radius \( r = 8 \) cm
- Height \( h = 15 \) cm
-
Dilation effect: When you dilate a shape by a scale factor \( k \), the new dimensions are given by: \[ \text{New radius} = k \times \text{Original radius} = 4 \times 8 , \text{cm} = 32 , \text{cm} \] \[ \text{New height} = k \times \text{Original height} = 4 \times 15 , \text{cm} = 60 , \text{cm} \]
-
Calculate the new surface area: The formula for the surface area \( SA \) of a cylinder is given by: \[ SA = 2\pi r h + 2\pi r^2 \] Plugging in the new values: \[ SA = 2\pi (32)(60) + 2\pi (32)^2 \]
Calculating each term:
- The lateral surface area: \[ 2\pi (32)(60) = 3840\pi \]
- The area of the two bases: \[ 2\pi (32)^2 = 2\pi (1024) = 2048\pi \]
-
Total surface area: Combining both parts: \[ SA = 3840\pi + 2048\pi = 5888\pi \]
-
Final result: Therefore, the new surface area of the dilated cylinder is: \[ SA = 5888\pi , \text{cm}^2 \approx 18479.7 , \text{cm}^2 , \text{(using } \pi \approx 3.14\text{)} \]
So, the new surface area of the cylinder after dilation is \( 5888\pi , \text{cm}^2 \).