6. A circular cylinder has radius r and height h. A second cylinder has radius 2r and height 2h. What is the volume of the second cylinder in terms of V = πr²h?
Answer: D
The volume of the second cylinder is 8V.
The volume of the second cylinder can be calculated using the formula for the volume of a cylinder, which is V = πr²h. Given that the second cylinder has a radius of 2r and a height of 2h, its volume becomes V = π(2r)²(2h) = π(4r²)(2h) = 8πr²h, which is 8 times the volume of the first cylinder (V = πr²h).
A) 2V
This option is incorrect because when calculating the volume of the second cylinder, we find that it is significantly larger than just double the volume of the first cylinder. The volume is determined by both the radius and height, both of which are doubled, leading to a much larger result.
B) 4V
While this option might seem plausible since the radius is doubled, it does not account for the doubling of height as well. The calculation shows that the volume actually increases by a factor of 8 due to both dimensions being multiplied, making this option incorrect.
C) 6V
This option is incorrect as it does not reflect the correct relationship between the dimensions of the two cylinders. The volume increase is not a simple addition, but rather a multiplicative effect from both the radius and height changes, leading to a total volume of 8V instead.
D) 8V
This is the correct option. The volume of the second cylinder, calculated with its doubled radius and height, results in a total volume that is 8 times greater than the first cylinder's volume, confirming that the correct answer is indeed 8V.
E) 16V
This option is incorrect because it overestimates the volume of the second cylinder. The relationship between dimensions does not lead to a volume that is 16 times greater; it is important to accurately account for both the increase in radius and height, which collectively result in only an 8-fold increase.
Conclusion
The correct answer is D) 8V, as the volume of the second cylinder is derived from the formula for cylinder volume, considering both the radius and height doubling. All other options fail to accurately reflect the mathematical relationship between the dimensions, which shows that the volume increases by a factor of 8, not 2, 4, 6, or 16.