Question

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 2 cm and the height of the cone is equal to its radius. The volume of the solid will be :

• A. $2\pi$ cm³
• B. $4\pi$ cm³
• C. $6\pi$ cm³
• D. $8\pi$ cm³

Hint:

When the height of a cone equals its radius ($h=r$), the volume of a cone plus a hemisphere with the same radius simplifies beautifully to just $\pi r^3$.

Answer 1

The Correct Option is D

To find the volume of the solid, which consists of a cone on top of a hemisphere with both having the same radius, let's start by breaking it down into two parts: the volume of the cone and the volume of the hemisphere.

1. The radius of both the cone and the hemisphere is given as 2 cm.
2. The height of the cone (h) is equal to its radius, which is also 2 cm.

1. Volume of the Cone:

The formula for the volume of a cone is: \(V_{\text{cone}} = \frac{1}{3} \pi r^2 h\)

• Substituting the given values: \(r = 2 \text{ cm}, \, h = 2 \text{ cm}\)
• \(V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (2) = \frac{1}{3} \pi \cdot 4 \cdot 2 = \frac{8}{3} \pi \, \text{cm}^3\)

2. Volume of the Hemisphere:

The formula for the volume of a hemisphere is: \(V_{\text{hemisphere}} = \frac{2}{3} \pi r^3\)

• Substituting the given value: \(r = 2 \text{ cm}\)
• \(V_{\text{hemisphere}} = \frac{2}{3} \pi (2)^3 = \frac{2}{3} \pi \cdot 8 = \frac{16}{3} \pi \, \text{cm}^3\)

Total Volume of the Solid:

To find the total volume of the solid, we add the volumes of the cone and the hemisphere:

• \(V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{8}{3} \pi + \frac{16}{3} \pi = \frac{24}{3} \pi = 8 \pi \, \text{cm}^3\)

Therefore, the volume of the solid is \(8\pi \, \text{cm}^3\).

The correct answer is: $8\pi$ cm³.