Question

A solid sphere is made by melting the three metallic solid spheres of diameters 12 cm, 16 cm and 20 cm. Find the diameter of this solid sphere.

Hint:

When melting spheres to form a new sphere, equate the total volume of the spheres to the volume of the new sphere and solve for the radius or diameter.

Answer 1

Let the radius of the new solid sphere be \( R \). The volume of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3. \]
Step 1: The volumes of the three metallic spheres are: - For the sphere with diameter 12 cm, the radius is \( r_1 = 6 \, \text{cm} \), so the volume is: \[ V_1 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = 288 \pi \, \text{cm}^3. \] - For the sphere with diameter 16 cm, the radius is \( r_2 = 8 \, \text{cm} \), so the volume is: \[ V_2 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi \times 512 = \frac{2048}{3} \pi \, \text{cm}^3. \] - For the sphere with diameter 20 cm, the radius is \( r_3 = 10 \, \text{cm} \), so the volume is: \[ V_3 = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi \times 1000 = \frac{4000}{3} \pi \, \text{cm}^3. \]
Step 2: The total volume of the three spheres is: \[ V_{\text{total}} = V_1 + V_2 + V_3 = 288 \pi + \frac{2048}{3} \pi + \frac{4000}{3} \pi. \] Simplify the total volume: \[ V_{\text{total}} = 288 \pi + \frac{2048 + 4000}{3} \pi = 288 \pi + \frac{6048}{3} \pi = 288 \pi + 2016 \pi = 2304 \pi \, \text{cm}^3. \]
Step 3: The volume of the new sphere is: \[ V_{\text{new}} = \frac{4}{3} \pi R^3. \] Equating the total volume of the spheres and the volume of the new sphere: \[ 2304 \pi = \frac{4}{3} \pi R^3. \] Cancel \( \pi \) from both sides: \[ 2304 = \frac{4}{3} R^3. \] Multiply both sides by 3: \[ 6912 = 4 R^3. \] Now divide both sides by 4: \[ R^3 = 1728 \quad \Rightarrow \quad R = \sqrt[3]{1728} = 12 \, \text{cm}. \]
Step 4: The diameter of the new solid sphere is \( 2R = 2 \times 12 = 24 \, \text{cm}. \)
Conclusion: The diameter of the new solid sphere is \( 24 \, \text{cm} \).