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

Step 1: Understanding the Concept:
The total volume of the solid is the sum of the volume of the cone and the volume of the hemisphere.

Step 2: Formulas and Values:

Radius ($r$) = 2 cm
Height of cone ($h$) = $r$ = 2 cm
Volume of Cone = $\frac{1}{3}\pi r^2 h$
Volume of Hemisphere = $\frac{2}{3}\pi r^3$
Step 3: Calculation:
$$\text{Total Volume} = \frac{1}{3}\pi r^2(r) + \frac{2}{3}\pi r^3 = \frac{1}{3}\pi r^3 + \frac{2}{3}\pi r^3$$
$$\text{Total Volume} = \pi r^3$$
Substituting $r = 2$:
$$\text{Total Volume} = \pi (2)^3 = 8\pi \text{ cm}^3$$
Step 4: Final Answer:
The volume of the solid is $8\pi$ cm³.