Volume of a hollow sphere is 11352/7 ​ cm^(3) . If the outer radius is 8cm, find the inner radius of the sphere. (Take π= 22/7)

Answer

To find the inner radius of the hollow sphere, we will use the formula for the volume of a sphere and the information given in the problem.

### Step-by-Step Solution:

1. **Understand the Volume of a Hollow Sphere**:
The volume \( V \) of a hollow sphere can be calculated using the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
where \( R \) is the outer radius and \( r \) is the inner radius.

2. **Substitute the Given Values**:
We know the outer radius \( R = 8 \) cm and the volume \( V = \frac{11352}{7} \) cm³. We will also use \( \pi = \frac{22}{7} \).

3. **Set Up the Equation**:
Plugging the values into the volume formula:
\[ \frac{4}{3} \times \frac{22}{7} \times (8^3 - r^3) = \frac{11352}{7} \]

4. **Simplify the Equation**:
First, calculate \( 8^3 \):
\[ 8^3 = 512 \]
Now substitute this value into the equation:
\[ \frac{4}{3} \times \frac{22}{7} \times (512 - r^3) = \frac{11352}{7} \]

5. **Eliminate the Fraction**:
Multiply both sides by \( 7 \) to eliminate the denominator:
\[ \frac{4}{3} \times 22 \times (512 - r^3) = 11352 \]

6. **Calculate \( \frac{4}{3} \times 22 \)**:
\[ \frac{4 \times 22}{3} = \frac{88}{3} \]
Now the equation becomes:
\[ \frac{88}{3} \times (512 - r^3) = 11352 \]

7. **Multiply Both Sides by 3**:
\[ 88 \times (512 - r^3) = 34056 \]

8. **Divide by 88**:
\[ 512 - r^3 = \frac{34056}{88} \]
Calculate \( \frac{34056}{88} = 387 \):
\[ 512 - r^3 = 387 \]

9. **Solve for \( r^3 \)**:
Rearranging gives:
\[ r^3 = 512 - 387 = 125 \]

10. **Find the Inner Radius \( r \)**:
Taking the cube root:
\[ r = \sqrt[3]{125} = 5 \text{ cm} \]

### Final Answer:
The inner radius of the hollow sphere is **5 cm**.