A sphere is melted and half of the molten liquid is used to form 11 identical cubes, whereas the remaining half is used to form 7 identical smaller spheres. The ratio of the side of the cube to the radius of the new small sphere is

Options

A. `(4//3)^(1//3)`
B. `(8//3)^(1//3)`
C. `(3)^(1//3)`
D. 2

Answer

To solve the problem step by step, we will follow the reasoning given in the video transcript.

### Step 1: Understand the volume of the original sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the original sphere.

### Step 2: Calculate the volume of half of the sphere
Since half of the sphere is used to form the cubes, the volume of the half-sphere is:
\[ \text{Volume of half-sphere} = \frac{1}{2} \times \frac{4}{3} \pi R^3 = \frac{2}{3} \pi R^3 \]

### Step 3: Volume of the cubes
Let \( A \) be the side of one cube. The volume of one cube is given by:
\[ \text{Volume of one cube} = A^3 \]
Since there are 11 identical cubes, the total volume of the cubes is:
\[ \text{Total volume of cubes} = 11 A^3 \]
Setting the volume of half the sphere equal to the total volume of the cubes gives us:
\[ \frac{2}{3} \pi R^3 = 11 A^3 \]

### Step 4: Rearranging to find \( A^3 \)
From the equation above, we can express \( A^3 \) in terms of \( R \):
\[ A^3 = \frac{2 \pi R^3}{33} \]

### Step 5: Volume of the smaller spheres
The remaining half of the sphere is used to form 7 smaller spheres. The volume of one smaller sphere (with radius \( r \)) is:
\[ \text{Volume of one smaller sphere} = \frac{4}{3} \pi r^3 \]
Thus, the total volume of the 7 smaller spheres is:
\[ \text{Total volume of smaller spheres} = 7 \times \frac{4}{3} \pi r^3 = \frac{28}{3} \pi r^3 \]
Setting this equal to the volume of the remaining half of the sphere gives us:
\[ \frac{2}{3} \pi R^3 = \frac{28}{3} \pi r^3 \]

### Step 6: Rearranging to find \( r^3 \)
From the equation above, we can express \( r^3 \) in terms of \( R \):
\[ r^3 = \frac{R^3}{14} \]

### Step 7: Finding the ratio \( \frac{A}{r} \)
Now we have expressions for \( A^3 \) and \( r^3 \):
1. \( A^3 = \frac{2 \pi R^3}{33} \)
2. \( r^3 = \frac{R^3}{14} \)

To find the ratio \( \frac{A}{r} \), we first find \( \frac{A^3}{r^3} \):
\[ \frac{A^3}{r^3} = \frac{\frac{2 \pi R^3}{33}}{\frac{R^3}{14}} = \frac{2 \pi}{33} \times \frac{14}{1} = \frac{28 \pi}{33} \]

Taking the cube root gives us:
\[ \frac{A}{r} = \left(\frac{28 \pi}{33}\right)^{1/3} \]

### Step 8: Final Ratio
To find the ratio of the side of the cube to the radius of the smaller sphere:
\[ \frac{A}{r} = \frac{A^3}{r^3}^{1/3} = \left(\frac{28}{33}\right)^{1/3} \]

### Conclusion
The ratio of the side of the cube to the radius of the small sphere is:
\[ \frac{A}{r} = \frac{2}{3} \]