The Ratio of Volumes of Two Spheres with Doubled Radius

To find the ratio of the volumes of two spheres when the radius of one sphere is doubled, we need to understand the relationship between the volume of a sphere and its radius. The volume of a sphere is given by the formula:

V = (4/3)πr³

where V is the volume and r is the radius of the sphere.

Step 1: Define the radius of the first sphere
Let's assume the radius of the first sphere is 'r'.

Step 2: Calculate the volume of the first sphere
Using the formula for the volume of a sphere, we can calculate the volume of the first sphere:

V₁ = (4/3)πr³

Step 3: Define the radius of the second sphere
Since the radius of the second sphere is double that of the first sphere, we can express it as:

r₂ = 2r

where r₂ is the radius of the second sphere.

Step 4: Calculate the volume of the second sphere
Using the formula for the volume of a sphere, we can calculate the volume of the second sphere:

V₂ = (4/3)π(2r)³
= (4/3)π8r³
= 8(4/3)πr³
= 8V₁

Step 5: Calculate the ratio of the volumes
To find the ratio of the volumes of the two spheres, we divide the volume of the second sphere by the volume of the first sphere:

Ratio = V₂ / V₁
= 8V₁ / V₁
= 8

Final Answer:
The ratio of the volumes of the two spheres, when the radius of the first sphere is doubled, is 8. This means that the volume of the second sphere is eight times greater than the volume of the first sphere.