17. A first circular cylinder has radius r and height h, with volume V = πr²h. A second cylinder has radius 2r and height 2h. Express its volume in terms of V.
Answer: D
The volume of the second cylinder is 8V.
To determine the volume of the second cylinder with radius 2r and height 2h, we apply the volume formula for a cylinder, which is V = πr²h. Substituting the new dimensions results in a volume of V = π(2r)²(2h) = π(4r²)(2h) = 8πr²h, which is 8 times the volume of the first cylinder, V.
A) 2V
This option suggests that the volume of the second cylinder is double that of the first. However, substituting the dimensions into the volume formula shows that the second cylinder's volume is significantly larger, thus making this option incorrect.
B) 4V
While this option indicates an increase in volume, it underestimates the growth due to both the radius and height being doubled. The actual calculation shows the volume to be 8V, so this option is also incorrect.
C) 6V
This choice proposes an increase to six times the original volume. However, based on the calculations from the cylinder's dimensions, this option does not reflect the correct multiplication of the original volume, rendering it incorrect.
D) 8V
This is the correct option, as the volume of the second cylinder calculated with the new dimensions is indeed 8 times that of the first cylinder. The formula confirms this outcome, making this the right answer.
E) 16V
This option suggests an even larger volume than calculated. The volume of the second cylinder does not reach this figure, as the calculations show it is only 8V, thus this option is incorrect.
Conclusion
The volume of the second cylinder is definitively calculated to be 8 times the volume of the first cylinder, confirming that option D is correct. All other options fail to accurately reflect the proportional increase in volume resulting from the doubling of both radius and height.