A solid metallic sphere of radius 4 cm is melted and recast into '4' identical cubes. What is the side of the cube?

Options

A. `7^(3)sqrt((pi)/(3))cm`
B. `5^(3)sqrt((pi)/(3))cm`
C. `6^(3)sqrt((pi)/(3))cm`
D. `4^(3)sqrt((pi)/(3))cm`

Answer

To find the side of the cube formed by melting a solid metallic sphere of radius 4 cm, we can follow these steps:

### Step 1: Calculate the Volume of the Sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given the radius \( r = 4 \) cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (4)^3 \]

### Step 2: Calculate \( (4)^3 \)
Calculating \( (4)^3 \):
\[ (4)^3 = 64 \]
So, substituting this back into the volume formula:
\[ V = \frac{4}{3} \pi (64) \]
\[ V = \frac{256}{3} \pi \text{ cm}^3 \]

### Step 3: Set the Volume of the Sphere Equal to the Volume of the Cubes
Since the sphere is melted and recast into 4 identical cubes, the total volume of the cubes will be equal to the volume of the sphere:
\[ V_{\text{cubes}} = 4 \times a^3 \]
where \( a \) is the side length of one cube.

Setting the volumes equal:
\[ \frac{256}{3} \pi = 4 \times a^3 \]

### Step 4: Solve for \( a^3 \)
To find \( a^3 \), we can divide both sides by 4:
\[ a^3 = \frac{256}{3 \times 4} \pi \]
\[ a^3 = \frac{256}{12} \pi \]
\[ a^3 = \frac{64}{3} \pi \]

### Step 5: Find the Side Length \( a \)
To find \( a \), we take the cube root of both sides:
\[ a = \sqrt[3]{\frac{64}{3} \pi} \]

### Step 6: Approximate the Value
Using the approximation \( \pi \approx 3.14 \):
\[ a \approx \sqrt[3]{\frac{64 \times 3.14}{3}} \approx \sqrt[3]{67.2} \]
Calculating the cube root gives us:
\[ a \approx 4.03 \text{ cm} \]

### Final Answer
The side of each cube is approximately \( 4.03 \) cm.

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