The height and the slant height of a cone are 21 cm and 28 cm respectively, the volume of the cone will be :

Question

Accepted Answer

To determine the volume of the cone, we need to use the formula for the volume of a cone, which is:

$$ V = \frac{1}{3} \pi r^2 h $$

where $ V $ is the volume, $ r $ is the radius of the base, and $ h $ is the height of the cone.

Given:
- Height ($ h $) = 21 cm
- Slant height = 28 cm

First, we need to find the radius ($ r $) of the base of the cone. We can use the Pythagorean theorem in the right triangle formed by the height, radius, and slant height of the cone:

$$ l^2 = r^2 + h^2 $$

where $ l $ is the slant height.

Substituting the given values:

$$ 28^2 = r^2 + 21^2 $$

$$ 784 = r^2 + 441 $$

$$ r^2 = 784 - 441 $$

$$ r^2 = 343 $$

$$ r = \sqrt{343} $$

Now, we can calculate the volume of the cone:

$$ V = \frac{1}{3} \pi (343) \times 21 $$

$$ V = \frac{1}{3} \times \pi \times 7203 $$

$$ V = 2401 \pi $$

Approximating $\pi$ as 3.14:

$$ V \approx 2401 \times 3.14 $$

$$ V \approx 7545.14 \, \text{cm}^3 $$

Rounding to the nearest whole number, the volume is approximately 7546 cm³.

Therefore, Option 3 (7546 cm³) is the correct answer.

**Explanation for Incorrect Options:**

1. **Option 1 (7456 cm³):** This value is incorrect because it does not match the calculated volume of approximately 7546 cm³. The discrepancy arises from incorrect calculations or assumptions.

2. **Option 2 (7654 cm³):** This value is incorrect as it exceeds the calculated volume of approximately 7546 cm³. It likely results from an overestimation in the calculation process.

3. **Option 4 (7746 cm³):** This value is incorrect because it is significantly higher than the calculated volume of approximately 7546 cm³. This option reflects an overestimation of the cone's volume.

In conclusion, the correct volume of the cone, based on the given dimensions and calculations, is best represented by Option 3 (7546 cm³).