Two similar cones have radii 2:3 . Find the ratios of their volumes?

Answer 1

To find the ratios of the volumes of two similar cones with radii 2:3, we need to consider that volume is directly proportional to the cube of the radius.

Let's assume that one cone has a radius of 2 units and another cone has a radius of 3 units.

The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where V represents volume, π is Pi (approximately 3.14), r stands for radius, and h indicates height.

Since both cones are assumed to be similar, we can keep their heights constant. For simplicity, let's say their height is equal to 'h' units.

Now we can calculate the ratio between their volumes:

Volume_1 / Volume_2 = [(1/3) * π * r_1^2 * h] / [(1/3) * π * r_2^2 * h]

The factors involving height cancel out due to similarity and division by itself results in 1:

Volume_1 / Volume_2 = [r_1^2] / [r_2^2]

Plugging in our given radii values:

Volume_1 / Volume_2 = [4] / [9]

Thus, the ratio between their volumes would be 4:9 or simply expressed as 4/9.

Therefore, if one cone has a volume represented by x cubic units, then other similar cone will have a volume equal to (4x)/9 cubic units.