Final answer:
When the radius of a cone is halved and its height is doubled, the volume of the cone is reduced to half of its original volume. This is because the base area becomes one fourth, and the height is doubled, ultimately resulting in a factor of 1/2 when calculating with the cone volume formula.
Step-by-step explanation:
The volume of a cone is given by the formula V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height. If the radius is halved, the new radius becomes r/2, and squaring this gives us (r/2)^2 = r^2/4, meaning the new base area is one fourth of the original. If the height is doubled, the new height becomes 2h. Putting these changes into the formula, we get the new volume V' = (1/3)π(r/2)^2(2h) = (1/4)(2)V = V/2. This shows us that the new volume is half the original volume, so the correct answer to the question is that the volume is halved.