The volume of the cylinder is ⅔ times larger than that of the sphere that fits perfectly inside it, as the cylinder's volume is 2πr³ and the sphere's volume is ⅔πr³. The ratio is constant, independent of size.
The question is asking how much larger a cylinder's volume is compared to a sphere that fits perfectly inside it.
To answer this question, we'll need to use the formulas for the volume of a cylinder V = πr²h and the volume of a sphere V = ⅔πr³.
When a sphere fits perfectly inside a cylinder, the height of the cylinder h is equal to the diameter of the sphere, which is 2r.
Therefore, the volume of the cylinder becomes V = πr²(2r) = 2πr³. The volume of the sphere is V = ⅔πr³.
To find out how much larger the cylinder is than the sphere, we subtract the volume of the sphere from the volume of the cylinder:
• cylinder volume - sphere volume = 2πr³ - ⅔πr³ = ⅔πr³.
So, the cylinder's volume is ⅔ times larger than the sphere's volume. This ratio is constant regardless of the actual size of the sphere or the cylinder.