Question : A metallic sphere of radius 10 cm is melted and then recast into small cones each of radius 3.5 cm and height 3 cm. The number of cones thus formed is:
Option 1: 140
Option 2: 132
Option 3: 112
Option 4: 126
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Correct Answer: 126
Solution :
The volume of a sphere
$V_{\text{s }} = \frac{4}{3} \pi r^3$ where $r$ is the radius.
The volume of a cone
$V_{\text{c}} = \frac{1}{3} \pi r^2 h$ where $r$ is the radius and $h$ is the height.
Given that the sphere and the cones are made of the same material.
Let the number of cones as $n$.
$V_{\text{s}} = n \times V_{\text{c}}$
⇒ $\frac{4}{3} \pi (10.5)^3 = n \times \frac{1}{3} \pi (3.5)^2 \times 3$
⇒ $n = \frac{4 \times (10.5)^3}{(3.5)^2 \times 3} = 126$
Hence, the correct answer is 126.
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