Weight of a solid metallic sphere of radius 4 cm is 4 kg. The weight of a hollow sphere made with same metal, whose outer diameter is 16 cm and inner diameter is 12 cm, is

Options

A. 20.5 kg
B. 15.5 kg
C. 16.5 kg
D. 18.5 kg

Answer

To solve the problem of finding the weight of the hollow sphere made from the same metal as the solid sphere, we will follow these steps:

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

Given that the radius of the solid sphere is 4 cm:
\[ V_{\text{solid}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the weight per cubic centimeter of the metal
The weight of the solid sphere is given as 4 kg. Therefore, the weight per cubic centimeter can be calculated as:
\[ \text{Weight per cm}^3 = \frac{\text{Weight}}{\text{Volume}} = \frac{4 \, \text{kg}}{\frac{256}{3} \pi \, \text{cm}^3} = \frac{4 \cdot 3}{256 \pi} = \frac{12}{256 \pi} \, \text{kg/cm}^3 \]

### Step 3: Find the volume of the hollow sphere
The hollow sphere has an outer diameter of 16 cm and an inner diameter of 12 cm. Thus, the outer radius \( R \) is:
\[ R = \frac{16}{2} = 8 \, \text{cm} \]
and the inner radius \( r \) is:
\[ r = \frac{12}{2} = 6 \, \text{cm} \]

The volume \( V \) of the hollow sphere is given by the difference between the volume of the outer sphere and the inner sphere:
\[ V_{\text{hollow}} = V_{\text{outer}} - V_{\text{inner}} = \left(\frac{4}{3} \pi R^3\right) - \left(\frac{4}{3} \pi r^3\right) \]
Substituting the values:
\[ V_{\text{hollow}} = \frac{4}{3} \pi (8^3) - \frac{4}{3} \pi (6^3) = \frac{4}{3} \pi (512 - 216) = \frac{4}{3} \pi (296) = \frac{1184}{3} \pi \, \text{cm}^3 \]

### Step 4: Calculate the weight of the hollow sphere
Now, we can find the weight of the hollow sphere using the weight per cubic centimeter:
\[ \text{Weight of hollow sphere} = \text{Volume} \times \text{Weight per cm}^3 \]
\[ = \frac{1184}{3} \pi \cdot \frac{12}{256 \pi} \]
The \( \pi \) cancels out:
\[ = \frac{1184 \cdot 12}{3 \cdot 256} = \frac{14112}{768} = 18.5 \, \text{kg} \]

### Final Answer:
The weight of the hollow sphere is **18.5 kg**.

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