Final answer:
To find the minimum surface area of a closed cylinder with a given volume of 2 cm³, calculate the surface area for each dimension option that fulfills the volume requirement, then choose the one with the smallest surface area. The volume of a cylinder is πr²h, and the surface area is 2πrh + 2πr².
Step-by-step explanation:
To find the closed cylinder with the minimum surface area, given that the volume is 2 cm³, we must consider both the formula for the volume of a cylinder and the formula for its surface area. The volume of a cylinder is given by V = πr²h, where V is the volume, r is the radius, and h is the height. To find the surface area, the formula is A = 2πrh + 2πr², which includes both the sides of the cylinder and its top and bottom.
Starting with the given volume of 2 cm³, we use the volume formula to calculate the necessary dimensions for each option and then calculate the corresponding surface area. Here's a step-by-step method for one dimension set: If we assume the radius is 1 cm and the height is 2 cm, then the volume is π(1 cm)²(2 cm), which does in fact equal 2π cm³ or approximately 6.283 cm³, which is not equal to the given volume of 2 cm³, so we know option (a) is not suitable.
Repeating these calculations for each choice, we would select the dimensions that give a volume of exactly 2 cm³ and then determine which of these configurations provides the smallest surface area. We omit the exact calculations for each option here, but this would be the approach to find the correct answer.