Final answer:
The surface area of a cylinder, given a fixed volume of 18 cubic units, is not simply expressed by one of the options provided. It is a combination of the area of the circular ends and the side surface area, which results in a formula that includes the term 2r² plus a term with 36 / r.
Step-by-step explanation:
To answer your question on the surface area S e of a cylinder with a given volume of 18 cubic units as a function of its radius r, we first need to recall the formula for the volume of a cylinder: V = πr²h, where V is the volume, π is Pi (approximately 3.14159), r is the radius, and h is the height. Our goal is to express the surface area in terms of the radius r, while keeping the volume constant at 18 cubic units.
The surface area S e of a cylinder can be calculated using the formula: S e = 2πr(h + r), which is the sum of the areas of the two circular ends (2πr²) and the side surface area (2πrh). Since we know the volume V = 18, we can express the height h in terms of r as h = V / (πr²). Substituting this into the surface area formula gives us S e = 2πr^2 + 36 / r, which is not one of the options you provided (a) 18r, (b) 2r², (c) 3r, or (d) 4r². Therefore, the correct expression for the surface area as a function of radius when the volume is fixed at 18 cubic units would be a combination of 2r² and a term involving 36 / r.