The volume of a sphere is `4851cm^(3)`. How much should it radius be reduced so that it's volume becomes `(4312)/(3)cm^(3)`?

Answer

To solve the problem of how much the radius of a sphere should be reduced so that its volume changes from \( 4851 \, \text{cm}^3 \) to \( \frac{4312}{3} \, \text{cm}^3 \), we can follow these steps:

### Step 1: Understand the Volume Formula
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 sphere.

### Step 2: Calculate the Original Radius
Given that the original volume \( V = 4851 \, \text{cm}^3 \), we can set up the equation:
\[ \frac{4}{3} \pi r^3 = 4851 \]
Using \( \pi \approx \frac{22}{7} \), we substitute this into the equation:
\[ \frac{4}{3} \cdot \frac{22}{7} \cdot r^3 = 4851 \]

### Step 3: Rearranging the Equation
To find \( r^3 \), we rearrange the equation:
\[ r^3 = \frac{4851 \cdot 3 \cdot 7}{4 \cdot 22} \]

### Step 4: Simplifying the Expression
Calculating the right-hand side:
1. First, simplify \( \frac{4851}{22} \):
- \( 4851 \div 11 = 441 \)
- \( 22 \div 11 = 2 \)
- Thus, \( \frac{4851}{22} = \frac{441}{2} \)

2. Now, substituting back:
\[ r^3 = \frac{441 \cdot 3 \cdot 7}{4 \cdot 2} \]
3. Simplifying further:
\[ r^3 = \frac{441 \cdot 21}{8} \]

### Step 5: Finding \( r \)
To find \( r \), we need to simplify \( r^3 \):
- \( 441 = 21^2 \)
- Therefore, \( r^3 = \frac{21^2 \cdot 21}{8} = \frac{21^3}{8} \)
- Taking the cube root gives:
\[ r = \frac{21}{2} = 10.5 \, \text{cm} \]

### Step 6: Calculate the New Radius
Now, we need to find the new radius \( r' \) for the new volume \( V' = \frac{4312}{3} \, \text{cm}^3 \):
\[ \frac{4}{3} \pi r'^3 = \frac{4312}{3} \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{4}{3} \cdot \frac{22}{7} \cdot r'^3 = \frac{4312}{3} \]

### Step 7: Rearranging for \( r'^3 \)
Rearranging gives:
\[ r'^3 = \frac{4312 \cdot 7}{4 \cdot 22} \]
Calculating \( \frac{4312}{22} \):
- \( 4312 \div 11 = 392 \)
- Thus, \( \frac{4312}{22} = \frac{392}{2} = 196 \)
- Therefore:
\[ r'^3 = \frac{196 \cdot 7}{4} \]

### Step 8: Simplifying Further
Calculating:
\[ r'^3 = \frac{1372}{4} = 343 \]
Taking the cube root gives:
\[ r' = 7 \, \text{cm} \]

### Step 9: Calculate the Reduction in Radius
Finally, the reduction in radius is:
\[ \text{Reduction} = r - r' = 10.5 \, \text{cm} - 7 \, \text{cm} = 3.5 \, \text{cm} \]

### Final Answer
The radius should be reduced by \( 3.5 \, \text{cm} \).

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