A hemisphere is inscribed in a cylinder and a cone is inscribed in the hemisphere .The vertex of cone lies on the centre of the upper circular part of the cylinder .Show that , `(1)/(3)xx`Volume of cylinder =`(1)/(2) xx`Volume of hemishphere= Volume of cone

Explanation

To prove that \( \frac{1}{3} \times \text{Volume of Cylinder} = \frac{1}{2} \times \text{Volume of Hemisphere} = \text{Volume of Cone} \), we will first derive the formulas for the volumes of the cylinder, hemisphere, and cone, and then show the relationships between these volumes.

### Step 1: Define the dimensions
Let the radius of the cylinder, hemisphere, and cone be \( r \). The height of the cylinder (which is also the height of the hemisphere) is \( h \).

### Step 2: Volume of the Cylinder
The volume \( V_c \) of a cylinder is given by the formula:
\[
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