A hollow cylindrical tube of 20 cm long is made of iron. If its outer and inner diameters are 8 cm and 6 cm respectively, then the volume of iron used in making the tube is :

Question

Accepted Answer

To determine the volume of iron used in making the hollow cylindrical tube, we need to calculate the volume of the material that forms the tube. This involves finding the difference between the volume of the outer cylinder and the volume of the inner cylinder.

1. **Calculate the volume of the outer cylinder:**
   - The outer diameter is 8 cm, so the outer radius (R) is 8/2 = 4 cm.
   - The height (h) of the cylinder is 20 cm.
   - The formula for the volume of a cylinder is $ V = \pi R^2 h $.
   - Therefore, the volume of the outer cylinder is:
     $$
     V_{\text{outer}} = \pi \times 4^2 \times 20 = \pi \times 16 \times 20 = 320\pi \, \text{cm}^3
     $$

2. **Calculate the volume of the inner cylinder:**
   - The inner diameter is 6 cm, so the inner radius (r) is 6/2 = 3 cm.
   - The height (h) remains the same at 20 cm.
   - The volume of the inner cylinder is:
     $$
     V_{\text{inner}} = \pi \times 3^2 \times 20 = \pi \times 9 \times 20 = 180\pi \, \text{cm}^3
     $$

3. **Calculate the volume of iron used:**
   - The volume of iron used is the difference between the outer and inner volumes:
     $$
     V_{\text{iron}} = V_{\text{outer}} - V_{\text{inner}} = 320\pi - 180\pi = 140\pi \, \text{cm}^3
     $$
   - Approximating $\pi$ as 3.14, we get:
     $$
     V_{\text{iron}} = 140 \times 3.14 = 439.6 \, \text{cm}^3
     $$
   - Rounding 439.6 to the nearest whole number gives us 440 cm$^3$.

Therefore, the volume of iron used in making the tube is approximately 440 cm$^3$, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (880 cm$^3$)**: This option is incorrect because it overestimates the volume of iron used. The calculation clearly shows that the volume difference between the outer and inner cylinders is much less than 880 cm$^3$.

- **Option 3 (1760 cm$^3$)**: This option is incorrect as it is significantly higher than the calculated volume. It likely represents a misunderstanding or miscalculation of the cylinder volumes.

- **Option 4 (240 cm$^3$)**: This option is incorrect because it underestimates the volume of iron used. The calculated volume of 440 cm$^3$ is nearly double this value, indicating that 240 cm$^3$ is not sufficient to account for the material used.

In conclusion, the correct answer is Option 1, as it accurately reflects the calculated volume of iron used in making the hollow cylindrical tube.