Final answer:
The minimum surface area for a cylinder with a volume of 500π cu. cm is achieved when the height is equal to the diameter. The volume formula, V = πr²h, must be used with the assumption that h = 2r to find the radius, and then calculate the surface area with SA = 2πr² + 2πrh.
Step-by-step explanation:
The student asked, "What is the minimum surface area possible for a right circular cylinder with a volume of 500π cu. cm?". To find this, we need to use the formula for the volume of a cylinder, which is V = πr²h, where V is the volume, r is the radius, and h is the height.
Since the volume is already given as 500π cu. cm, we must first consider that the surface area of a cylinder (SA) is given by SA = 2πr² + 2πrh, which includes both the area of the two circular ends and the area of the side surface. To determine the minimum surface area, we look for a cylinder where the height is equal to the diameter (or 2r), because a cylinder of these proportions is known to have the most efficient surface area to volume ratio.
The volume is V = πr²h = 500πcm³. Assuming h = 2r, we substitute h in the volume formula and solve for r, then calculate the surface area using the found radius and height. By doing this, we can determine the minimum surface area possible for the cylinder with the given volume.