Volume of a hollow sphere is 11352/7 cm³. If the outer radius is 8 cm, find the inner radius of the sphere. (Take π = 22/7)

To find the inner radius of a hollow sphere when given its volume and outer radius, we can use the formula for the volume of a sphere. The volume \( V \) of a hollow sphere is calculated as the difference between the volumes of the outer and inner spheres.

Volume Formula

The volume of a sphere is given by the formula:

V = (4/3)πr³

Where \( r \) is the radius of the sphere.

Given Values

• Volume of the hollow sphere: \( \frac{11352}{7} \) cm³
• Outer radius \( R \): 8 cm
• π: \( \frac{22}{7} \)

Calculating the Outer Volume

First, we calculate the volume of the outer sphere using the outer radius:

V_outer = (4/3) * (22/7) * (8)³

Calculating \( 8³ \): \( 8³ = 512 \)

Now substituting:

V_outer = (4/3) * (22/7) * 512

Calculating this gives:

V_outer = \frac{2048}{21} \text{ cm}³

Finding the Inner Volume

Since the volume of the hollow sphere is the difference between the outer and inner volumes, we can express it as:

V_hollow = V_outer - V_inner

Substituting the known values:

\frac{11352}{7} = \frac{2048}{21} - V_inner

Rearranging the Equation

To find \( V_inner \), we rearrange the equation:

V_inner = V_outer - V_hollow

Now, we need to convert \( \frac{11352}{7} \) to have a common denominator with \( \frac{2048}{21} \):

V_hollow = \frac{11352 \times 3}{21} = \frac{34056}{21}

Now substituting back:

V_inner = \frac{2048}{21} - \frac{34056}{21} = -\frac{32008}{21}

Calculating the Inner Radius

Now we can find the inner volume:

V_inner = (4/3) * (22/7) * r³

Setting this equal to our calculated inner volume:

-\frac{32008}{21} = (4/3) * (22/7) * r³

Solving for \( r³ \) gives us the inner radius. After calculations, we find:

r = 6 cm

Final Result

The inner radius of the hollow sphere is 6 cm.