Final answer:
None of the stated options about the radius or area of the base are correct, but the cylinder's height is indeed 4x units, based on the given volume and the formula for the volume of a cylinder.
Step-by-step explanation:
The question involves determining which statements are true about the given cylinder using the volume and dimensions provided. Since we know the volume V of the cylinder is πx^3 cubic units, and the volume formula for a cylinder is V = πr^2h, we can compare these formulas to find relationships between r, h, and x. Here are the evaluations: a. The radius of the cylinder is 2x units. This is false because, if the diameter is x units, the radius would be x/2 units. b. The area of the cylinder’s base is one-fourth πx^2 square units. This is also false because the area of the base (which is a circle) is πr^2 = π(x/2)^2 = πx^2/4, which is not one-fourth πx^2 but rather πx^2 divided by 4. c. The area of the cylinder’s base is one-half πx^2 square units. This is false; as explained above, it would actually be πx^2/4. d. The height of the cylinder is 2x units. To find the height, we compare the given volume equation with the volume formula for a cylinder: πx^3 = π(x/2)^2h, which simplifies to x^3 = (x^2/4)h. Solving for h gives us h = 4x; therefore, this option is also false. e. The height of the cylinder is 4x units. Based on the calculations above, this statement is true. We correctly derived that the height h = 4x when comparing the volume formulas Therefore, the correct statements about the cylinder are none of the given options hold true regarding the radius and the area of the base, but option e regarding the height is correct.