Suppose a cylinder and a cone share the same radius and the same height, which is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a cylinder?
To find the ratio between the volumes of a cone and a cylinder, we need to compare their formulas:
The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height.
The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height.
Since the height of the cone is twice the radius, we can substitute 2r for h in both formulas:
For the cylinder, V = πr²(2r) = 2πr³
For the cone, V = (1/3)πr²(2r) = (2/3)πr³
The ratio between the volumes of the cone and the cylinder is (2/3)πr³ / 2πr³.
The πr³ cancels out, leaving us with the ratio (2/3) / 2, which simplifies to 1/3.
Therefore, the ratio that describes the relationship between the volume formulas for the cone and the cylinder is 1:3.