A sphere of radius 7 cm is melted and recast into small spheres of radius 3.5 cm each. How many such spheres can be made?

Options

A. 4
B. 2
C. 9
D. 8

Answer

To solve the problem of how many small spheres can be made from a larger sphere that has been melted, we will follow these steps:

### Step 1: Calculate the Volume of the Larger Sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the larger sphere, the radius \( r \) is 7 cm. Thus, the volume \( V_L \) of the larger sphere is:
\[ V_L = \frac{4}{3} \pi (7)^3 \]

### Step 2: Calculate the Volume of the Smaller Sphere
Using the same volume formula for the smaller sphere, where the radius \( r \) is 3.5 cm, we have:
\[ V_S = \frac{4}{3} \pi (3.5)^3 \]

### Step 3: Set Up the Equation
Since the larger sphere is melted and recast into smaller spheres, the volume of the larger sphere will equal the total volume of the smaller spheres. If \( n \) is the number of smaller spheres, we can write:
\[ V_L = n \cdot V_S \]

### Step 4: Substitute the Volumes
Substituting the volumes we calculated:
\[ \frac{4}{3} \pi (7)^3 = n \cdot \frac{4}{3} \pi (3.5)^3 \]

### Step 5: Simplify the Equation
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ (7)^3 = n \cdot (3.5)^3 \]

### Step 6: Calculate the Cubes
Calculating the cubes:
\[ 7^3 = 343 \quad \text{and} \quad (3.5)^3 = 42.875 \]

### Step 7: Solve for \( n \)
Now substituting the values:
\[ 343 = n \cdot 42.875 \]
To find \( n \):
\[ n = \frac{343}{42.875} \]
Calculating this gives:
\[ n = 8 \]

### Conclusion
Thus, the number of smaller spheres that can be made is \( n = 8 \).