Question

Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere ?

Answer 1

Let the radius of the bigger sphere be x cm.
We know that

\[Density = \frac{Mass}{Volume}\]

As the spheres are made of the same metal, their densities will be equal.

\[\Rightarrow \frac{1}{\frac{4}{3}\pi \times 3^3} = \frac{7}{\frac{4}{3}\pi \times x^3}\]
\[ \Rightarrow x^3 = 3^3 \times 7 . . . . . \left( 1 \right)\]

Two spheres are melted to form a single big sphere.

\[\therefore \frac{4}{3}\pi \left( 3 \right)^3 + \frac{4}{3}\pi \left( x \right)^3 = \frac{4}{3}\pi R^3 \]
\[ \Rightarrow \left[ 3^3 + x^3 \right] = R^3 \]
\[ \Rightarrow \left[ 3^3 + 3^3 \times 7 \right] = R^3 \left[ Using \left( 1 \right) \right]\]
\[ \Rightarrow \left[ 3^3 \left( 1 + 7 \right) \right] = R^3 \]
\[ \Rightarrow \left[ 3^3 \times 8 \right] = R^3 \]
\[ \Rightarrow 3^3 \times 2^3 = R^3 \]
\[ \Rightarrow R = 3 \times 2 = 6 cm\]

∴ Diameter

\[= 2R = 2 \times 6 = 12 cm\]

Hence, the diameter of the new sphere is 12 cm.