Question

Consider a right cone with radius 2 and height 6. Its volume is V = π(2)26, or 8π units3.


If the height is changed to 3, does this have the same effect on the volume as changing the radius to 1?

If the height is now 3, then the new volume is
π units3.

If the radius is now 1, then the new volume is
π units3.

Therefore, changing the height to half of its original value and changing the radius to half of its original value does
the volume. Halving the height of the cone
the volume, while halving the radius of the cone results in
the volume.

Answer 1

First check the original volume:
V = (1/3)π r^2 h = (1/3)π(2)^2(6) = (1/3)π·4·6 = 8π.

If the height is 3 (half), with r = 2:
V = (1/3)π(2)^2(3) = (1/3)π·4·3 = 4π units^3.

If the radius is 1 (half), with h = 6:
V = (1/3)π(1)^2(6) = (1/3)π·1·6 = 2π units^3.

So halving the height halves the volume (8π → 4π), while halving the radius reduces the volume to one quarter (8π → 2π). They do not have the same effect. This follows from V ∝ h and V ∝ r^2.