Introduction
To find the radius of the new sphere formed by melting two smaller spheres, we will first determine the volumes of the original spheres and then use that information to calculate the radius of the larger sphere.
Volume of the Original Spheres
- The volume \( V \) of a sphere is calculated using the formula:
V = (4/3) * π * r³
- For Sphere 1 (1 kg, radius 3 cm):
- Radius (r1) = 3 cm
- Volume (V1) = (4/3) * π * (3)³ = 36π cm³
- For Sphere 2 (3 kg):
- Density of Sphere 2 can be calculated as Density = Mass/Volume
- The total mass of both spheres = 1 kg + 3 kg = 4 kg
- Since the total volume of the new sphere is the sum of the volumes of the original spheres, we can find the volume of Sphere 2 by using its mass.
Volume Calculation for Sphere 2
- Assuming the density of the material is uniform, the volume can be calculated as:
- Density = Mass/Volume
- Let the density be "d".
- For Sphere 2:
Mass = 3 kg = 3000 g
V2 = Mass/density = 3000/g cm³ (since we need density in grams/cm³)
- Assuming both spheres are made of the same material, we can use the density of Sphere 1 to find the volume of Sphere 2.
Total Volume of the New Sphere
- Total Volume (V_total) = V1 + V2
- V_total = 36π + (3000/d) cm³
Radius of the New Sphere
- The radius (R) of the new sphere can be calculated using the total volume:
- V_total = (4/3) * π * R³
- Rearranging gives:
R³ = (3 * V_total)/(4π)
Now, substitute the values to find R. The radius of the newly formed sphere can be calculated based on the total volume obtained from the original spheres.