Rationale
The volume of the second cylinder is 8V.
The volume of a cylinder is calculated using the formula V = πr²h. For the second cylinder with a radius of 2r and height of 2h, substituting these values into the formula yields a volume of 8V.
A) The first option suggests the volume is 2V, which would only be true if both the radius and height were halved. However, since the second cylinder's dimensions are both doubled, this option underestimates the volume significantly.
B) An answer of 4V implies that the volume increases by a factor of 4, which would be the case if only one of the dimensions (either the radius or height) was doubled. Since both dimensions are increased, this option incorrectly calculates the volume.
C) The choice of 6V suggests a misunderstanding of volume scaling. This answer does not take into account the correct scaling for both dimensions, as the volume increases with the cube of the scaling factor when both dimensions are doubled.
D) This is the correct answer, as it accurately reflects the volume of the second cylinder. The calculations show that when both the radius and height are doubled, the volume increases by a factor of 8, calculated as (2r)² * (2h) = 4r² * 2h = 8πr²h = 8V.
E) The final option of 16V mistakenly assumes an even greater increase in volume. This error arises from incorrectly applying the scaling, likely treating the changes in both dimensions as multiplicative factors rather than squared for the radius and linear for the height.
Conclusion
The volume of the second cylinder, derived from its dimensions, is accurately calculated to be 8V, demonstrating the relationship between changes in dimensions and the resulting volume. The scaling factors for both radius and height lead to a volume increase by a factor of 2³ = 8, confirming that V = π(2r)²(2h) results in 8V, illustrating the geometric principles at play in cylindrical volume calculations.