Question : A solid metallic sphere of radius 6.3 cm is melted and recast into a right circular cone of height 25.2 cm. What is the ratio of the diameter of the base to the height of the cone?
Option 1: 2 : 1
Option 2: 3 : 2
Option 3: 1 : 2
Option 4: 2 : 3
Correct Answer: 1 : 2
Solution :
The radius of the sphere ($R$) = 6.3 cm
The height of the cone ($h$) = 25.2 cm
Volume of sphere = $\frac{4}{3} \pi R^3$
Volume of cone = $\frac{1}{3} \pi r^2h$
According to the question
$\frac{4}{3} \pi R^3$ = $\frac{1}{3} \pi r^2 h$
⇒ ($\frac{4}{3}$) × 6.3 × 6.3 × 6.3 = $\frac{1}{3} \times r^2 \times 25.2$
⇒ $r = \sqrt{\frac{4 \times 6.3 \times 6.3 \times 6.3}{25.2}}$
⇒ $r = 6.3$
Diameter = 2 × 6.3 = 12.6 cm
So, the required ratio = 12.6 : 25.2 = 1 : 2
Hence, the correct answer is 1 : 2.
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