Question : The diameter of a sphere is twice the diameter of another sphere. The curved surface area of the first and the volume of the second are numerically equal. The numerical value of the radius of the first sphere is:
Option 1: 3
Option 2: 24
Option 3: 8
Option 4: 16
Correct Answer: 24
Solution :
Let the radius of the first sphere be $r$ and the radius of the second sphere be $R$.
According to the question,
$2r=2R×2$
⇒ $R=\frac{r}{2}$
The curved surface area of the first sphere = $4\pi r^2$
The volume of the second sphere = $\frac{4}{3}\pi R^3$
According to the question,
$4\pi r^2=\frac{4}{3}\pi R^3$
⇒ $r^2=\frac{1}{3} (\frac{r}{2})^3$
⇒ $r^2=\frac{r^3}{24}$
$\therefore r =24$ cm
Hence, the correct answer is 24.
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