Question : Three spherical balls of radius 2 cm, 4 cm, and 6 cm are melted to form a new spherical ball. In this process, there is a loss of 25% of the material. What is the radius (in cm) of the new ball?
Option 1: 6
Option 2: 8
Option 3: 12
Option 4: 16
Correct Answer: 6
Solution :
Given: The radii of three spherical balls are 2 cm, 4cm, and 6cm.
Percentage loss of material during melting = 25%
Let the radius of the new ball be $R$ cm
So, Percentage of usable material = 75%
Volume of sphere = $\frac{4}{3}\pi r^3$
Total volume of three spheres = Volume lost + volume of new ball
Volume of new ball = $\frac{75}{100}\times (\frac{4}{3}\pi \times 2^3 + \frac{4}{3}\pi \times 4^3+\frac{4}{3}\pi \times 6^3)$
⇒ $\frac{4}{3}\pi R^3=\frac{3}{4}\times (\frac{4}{3}\pi \times 2^3 + \frac{4}{3}\pi \times 4^3+\frac{4}{3}\pi \times 6^3)$
⇒ R = $\sqrt[3]{\frac{3}{4}\times(8 + 64+216)}$
⇒ R = 6 cm
Hence, the correct answer is 6.
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