Question : Three solid metallic spheres of radii 1 cm, 6 cm, and 8 cm, respectively, are melted and recast into a single solid sphere. The radius of the new sphere formed is:
Option 1: 9.0 cm
Option 2: 5.9 cm
Option 3: 7.7 cm
Option 4: 8.5 cm
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Correct Answer: 9.0 cm
Solution :
Given: The three original spheres had radii of 1 cm, 6 cm, and 8 cm, respectively.
Volume of a sphere $= \frac{4}{3} \pi r^3$ (where $r$ is the radius)
The total volume of the three original spheres
$=\frac{4}{3} \pi ×1^3 + \frac{4}{3} \pi × 6^3 + \frac{4}{3} \pi × 8^3$
$= \frac{4}{3} (\pi+216 \pi + 512 \pi)$
$= \frac{4}{3} × 729 \pi \ \text{cm}^3$
Let the radius of the new sphere formed be $R$ cm.
The volume of the new sphere = Sum of the volumes of the original spheres
⇒ $\frac{4}{3} \pi ×R^3 = \frac{4}{3} \pi × 729$
⇒ $R^3 = 729$
⇒ $R = 9$
So, the radius of the new sphere formed is 9.0 cm.
Hence, the correct answer is 9.0 cm.
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