Question : If the diameter of a sphere is reduced to its half, then the volume would be:
Option 1: increased by $\frac{1}{4}$ of the former volume
Option 2: reduced by $\frac{1}{4}$ of the former volume
Option 3: increased by $\frac{1}{8}$ of the former volume
Option 4: reduced by $\frac{1}{8}$ of the former volume
Correct Answer: reduced by $\frac{1}{8}$ of the former volume
Solution :
Given,
The diameter of a sphere is reduced to its half.
⇒ Radius will also be reduced by half.
We know the Volume of the sphere is $\propto\ r^3$, where $r$ is the radius of the sphere.
⇒ $\frac{\text{New Volume}}{\text{Old Volume}}=\frac{(\frac{r}{2})^3}{r^3}$
⇒ $\frac{\text{New Volume}}{\text{Old Volume}}=\frac{\frac{r^3}{8}}{r^3}$
⇒ $\frac{\text{New Volume}}{\text{Old Volume}}=\frac18$
⇒ The new volume will be reduced by $\frac{1}{8}$ of the former volume
Hence, the correct answer is "reduced by $\frac{1}{8}$ of the former volume".
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