Q1:

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1$ cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.

Given:

1. A solid consists of a cone mounted on a hemisphere.
2. The radius of the hemisphere ($r$) = $1$ cm.
3. The radius of the base of the cone ($r$) = $1$ cm.
4. The height of the cone ($h$) = radius of the cone = $1$ cm.

To Find:

The total volume of the solid in terms of $\pi$.

Step 1: Formulae Identification

The total volume of the solid ($V_{total}$) is the sum of the volume of the cone ($V_{cone}$) and the volume of the hemisphere ($V_{hemisphere}$).

Formula for the volume of a cone: $V_{cone} = \frac{1}{3}\pi r^2 h$

Formula for the volume of a hemisphere: $V_{hemisphere} = \frac{2}{3}\pi r^3$

Step 2: Calculating the Volume of the Cone

Substitute $r = 1$ cm and $h = 1$ cm into the formula:

$V_{cone} = \frac{1}{3} \times \pi \times (1)^2 \times 1$

$V_{cone} = \frac{1}{3} \times \pi \times 1 \times 1$

$V_{cone} = \frac{\pi}{3} \text{ cm}^3$

Step 3: Calculating the Volume of the Hemisphere

Substitute $r = 1$ cm into the formula:

$V_{hemisphere} = \frac{2}{3} \times \pi \times (1)^3$

$V_{hemisphere} = \frac{2}{3} \times \pi \times 1$

$V_{hemisphere} = \frac{2\pi}{3} \text{ cm}^3$

Step 4: Calculating the Total Volume

$V_{total} = V_{cone} + V_{hemisphere}$

$V_{total} = \frac{\pi}{3} + \frac{2\pi}{3}$

$V_{total} = \frac{\pi + 2\pi}{3}$

$V_{total} = \frac{3\pi}{3}$

$V_{total} = \pi \text{ cm}^3$

Final Answer: The volume of the solid is $\pi \text{ cm}^3$.