The volume of spheres are proportional to the cubes of their radii. Two spheres of the same material weigh 3.6 kg and 2.7 kg and the radius of the smaller one is 2 cm. If the two were melted down and formed into a single sphere, what would be its radius?

Options

A. 4 cm
B. 4.3 cm
C. 3 cm
D. 2.6 cm

Answer

To find the radius of the new sphere formed by melting down two spheres of different weights, we can follow these steps:

### Step 1: Understand the relationship between mass and volume
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
The mass \( m \) of the sphere is proportional to its volume since they are made of the same material. Thus, we can say:
\[ m \propto r^3 \]

### Step 2: Set up the known values
We have:
- Mass of sphere 1, \( m_1 = 3.6 \, \text{kg} \)
- Mass of sphere 2, \( m_2 = 2.7 \, \text{kg} \)
- Radius of sphere 2 (the smaller sphere), \( r_2 = 2 \, \text{cm} \)

### Step 3: Find the mass ratio
The total mass of the new sphere formed by melting the two spheres is:
\[ m_3 = m_1 + m_2 = 3.6 \, \text{kg} + 2.7 \, \text{kg} = 6.3 \, \text{kg} \]

### Step 4: Use the mass ratio to find the radius of the new sphere
From the relationship \( m \propto r^3 \), we can express the radius of the new sphere \( r_3 \) in terms of the radius of the smaller sphere \( r_2 \):
\[ \frac{m_3}{m_2} = \frac{r_3^3}{r_2^3} \]
Rearranging gives us:
\[ r_3^3 = r_2^3 \cdot \frac{m_3}{m_2} \]

### Step 5: Substitute the known values
Substituting the known values into the equation:
\[ r_3^3 = (2 \, \text{cm})^3 \cdot \frac{6.3 \, \text{kg}}{2.7 \, \text{kg}} \]
Calculating \( (2 \, \text{cm})^3 \):
\[ (2 \, \text{cm})^3 = 8 \, \text{cm}^3 \]
Now substituting this into the equation:
\[ r_3^3 = 8 \, \text{cm}^3 \cdot \frac{6.3}{2.7} \]

### Step 6: Calculate the ratio
Calculating \( \frac{6.3}{2.7} \):
\[ \frac{6.3}{2.7} = 2.3333 \quad (\text{approximately}) \]
Now substituting back:
\[ r_3^3 = 8 \, \text{cm}^3 \cdot 2.3333 = 18.6664 \, \text{cm}^3 \]

### Step 7: Take the cube root to find \( r_3 \)
Now, take the cube root of \( r_3^3 \):
\[ r_3 = \sqrt[3]{18.6664} \approx 2.675 \, \text{cm} \]

### Final Answer
The radius of the new sphere formed by melting the two spheres is approximately:
\[ \boxed{2.675 \, \text{cm}} \]