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Solution:
Step 1: Understand the Problem
To solve this problem, we need to find the ratio of the volumes of two spheres when the radius of one sphere is doubled.
Step 2: Use the Volume Formula for a Sphere
The volume VV of a sphere is given by the formula:
V=43πr3V=34πr3
Where:
Step 3: Determine the Ratios
Let's denote:
Given that the radius of the second sphere is twice the radius of the first sphere, we have:
r2=2r1r2=2r1
Step 4: Calculate the Ratios
Substituting the values into the volume formula, we get:
For the first sphere: V1=43πr13V1=34πr13
For the second sphere: V2=43π(2r1)3V2=34π(2r1)3
Now, we can find the ratio of their volumes:
Ratio of volumes=V2V1=43π(2r1)343πr13Ratio of volumes=V1V2=34πr1334π(2r1)3
=8r13πr13π=r13π8r13π
=81=8=18=8
Step 5: Conclusion
The ratio of the volumes of the two spheres is 8:18:1.
So, when the radius of a sphere is doubled, the ratio of their volumes becomes 8:18:1.
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