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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