a. height: 36 units
b. radius: 4 units
The volume of a cone is given by the formula ...
V = 1/3πr²h . . . . . for radius r and height h
This tells you the volume is jointly proportional to the height and the square of the radius. This means that for a given volume, the height is inversely proportional to the square of the radius. We can use this proportionality to answer the questions here.
The radius of Cone B is 2 units. Compared to Cone A, the radius of Cone B is 2/6 = 1/3 of the radius of cone A. The square of this ratio is (1/3)² = 1/9, and the inverse of that square is 9/1. This means the height of Cone B is 9 times the height of Cone A, so is ...
Cone B height = 9 × 4 units = 36 units
The height of this Cone B is 9 units, whereas the height of Cone A is 4 units. This means the height of Cone B is 9/4 times the height of Cone A. This is the inverse of the square of the ratio of radii of Cone B to Cone A. Inverting the height ratio, we get 4/9. Taking the square root of that, we get √(4/9) = 2/3. This means the radius of Cone B is 2/3 the radius of Cone A:
Cone B radius = 2/3 × 6 units = 4 units
We can check the above dimensions using the volume formula.
a. V = 1/3π(2 units)²(36 units) = 48π units³
b. V = 1/3π(4 units)²(9 units) = 48π units³
These are the expected volume, so the answers check OK.