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Three spheres of radii 6 cm, 8 cm and 10 cm are melted to form a sphere with a radius of:

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11 cm

13 cm

24 cm

12 cm

answer is D.

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Detailed Solution

Given the three sphere with radii 6 cm, 8 cm and 10 cm are melted and made a new sphere.
Let

r_{1},

r_{2}, r_{3}

be the radii of the three spheres respectively.
The formula for the volume of sphere is given by,
Volume of sphere

= \frac{4}{3} π r^{3},

where r is its radius.
So, after substituting the value of all the dimensions in the formula we get,
Volume of first sphere =

\frac{4}{3} π {r_{1}}^{3}

= 43 × π × (6) 3 = 4 π 3 × 216 c m 3

Volume of second sphere=

\frac{4}{3} π {r_{2}}^{3}

= 43 × π × (8) 3 = 4 π 3 × 512 c m 3

And volume of third sphere=

\frac{4}{3} π {r_{3}}^{3}

= 43 × π × (10) 3 = 4 π 3 × 1000 c m 3

Then total volume of new sphere:
=

\frac{4 π}{3} \times 216 + \frac{4 π}{3} \times 512 + \frac{4 π}{3} \times 1000

= \frac{4 π}{3} {\times 1728 cm}^{3}

Let the radius of new sphere is R.
So,

\frac{4}{3} π R^{3} = \frac{4 π}{3} \times {1728 cm}^{3}

\Rightarrow R^{3} = 1728

\Rightarrow R = 12

cm
Therefore, the radius of the new sphere is 12cm.
Hence option (4) is correct.

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