Question : 24 equal solid hemispheres are melted to form a right circular cylinder of radius 12 cm and height 24 cm. Find the radius of each solid hemisphere.
Option 1: 4 cm
Option 2: 8 cm
Option 3: 6 cm
Option 4: 3 cm
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Correct Answer: 6 cm
Solution :
The total volume of the 24 solid hemispheres is equal to the volume of the cylinder.
The volume of the cylinder = $\pi r^2h$
Volume of the cylinder (height $h$, radius $r$) = $\pi(12)^2 × 24 = 144 × 24\pi$
Since the 24 solid hemispheres have the same volume as the cylinder, the volume of each hemisphere can be calculated as:
Now, equate the volume of 24 hemispheres (each having a radius equal to $R$) to the cylinder.
$⇒24 × (\frac{2}{3}) \pi R^3 = 144 × 24\pi$
$⇒\frac{2}{3}R^3 = 144$
$⇒R^3 = 216$
$⇒R = 6$
Therefore, the radius of each solid hemisphere is 6 cm.
Hence, the correct answer is 6 cm.
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