Final answer:
The dimensions that would make the volume of cone W 1/27 times the volume of cone T is a radius that is 1/3 of cone T's radius, making Option C) radius = 1/3 r the correct choice.
Step-by-step explanation:
The volume of a right circular cone (cone T) with radius r and height h is given by the formula V = (1/3)πr²h. To find the volume of cone W, which is 1/27 times the volume of cone T, we must determine the dimensions of cone W that will yield this ratio.
Considering the volume of a cone is proportional to the cube of its linear dimensions, we can deduce that if the volume of cone W is 1/27 of cone T, then each linear dimension of cone W must be 1/3 (since (1/3)^3 = 1/27) of the corresponding dimension in cone T.
Therefore, Option C) radius = 1/3 r is correct because a cone with a radius that is 1/3 of the radius of cone T will have a volume that is 1/27 of cone T's volume, when the height remains the same.
It is important to note that changing the height alone, as in Option A) and Option B), would not suffice because the volume varies with the square of the radius as well. Meanwhile, Option D) radius = 3r would result in a volume increase, not a decrease.