Final answer:
The volume of the sphere is one-third of the volume of the cylinder with the same radius and height. Given the cylinder's volume is 30 m³, the sphere's volume would therefore be 10 m³.
Step-by-step explanation:
A sphere and a cylinder with the same radius and height have different volumes. To find the volume of the sphere when the volume of the cylinder is known, we can use the formula for the volume of a cylinder V = πr²h and the formula for the volume of a sphere V = ⅔πr³. If the volume of the cylinder is 30 m³, and since the sphere's radius and height are equivalent to the cylinder's, the sphere's volume should be less since its volume is calculated by fractional multiples of πr³.
Since both the cylinder and sphere have the same radius and height, let's denote the common height and diameter as 'h' and '2r' respectively. Based on the volume of the cylinder, which is 30 m³, we can imply that πr²h = 30 m³. To find the volume of the sphere, we will use the formula V = ⅔πr³. In terms of height, the sphere's radius r is equal to h/2, substituting this into the sphere's volume formula gives V = ⅔π(h/2)³.
Executing this calculation, we find that the volume of the sphere ends up being one-third of the cylinder's volume, which means the sphere's volume is 30 m³ / 3 = 10 m³. Therefore, the correct answer is option a.