Final answer:
To obtain the new surface area after dilation of a cylinder with a scale factor of 3, the original surface area is calculated and then the scale is applied to the dimensions, resulting in a new area of 1890π mm².
Step-by-step explanation:
To find the new surface area of a cylinder after dilation with a scale factor of k=3, we first need to calculate the original surface area and then apply the scale factor. The formula for the surface area of a cylinder is 2πrh + 2πr², where r is the radius and h is the height. The original radius is half of the diameter, so r = 5 mm, and h = 16 mm.
The original surface area is therefore A = 2π(5 mm)(16 mm) + 2π(5 mm)² = 160π mm² + 50π mm² = 210π mm². When the cylinder is dilated by a factor of 3, both the radius and the height are multiplied by 3, so the new dimensions are r = 15 mm and h = 48 mm. The new surface area is now A = 2π(15 mm)(48 mm) + 2π(15 mm)², which simplifies to 1440π mm² + 450π mm² = 1890π mm², which corresponds to Option A.