From a solid cylinder of height 10 cm and radius of the base 6 cm, a cone of same height and same base is removed. The volume of the remaining solid is :

Options

A. `240 pi` cu. cm
B. 5280 cu. cm
C. `620pi` cu. cm
D. `360 pi` cu. cm

Answer

To find the volume of the remaining solid after removing a cone from a cylinder, we can follow these steps:

### Step 1: Calculate the volume of the cylinder
The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.

Given:
- Radius \( r = 6 \) cm,
- Height \( h = 10 \) cm.

Substituting the values into the formula:

\[ V_{cylinder} = \pi (6^2)(10) = \pi (36)(10) = 360\pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of the cone
The formula for the volume \( V \) of a cone is given by:

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

Using the same radius and height as the cylinder:

\[ V_{cone} = \frac{1}{3} \pi (6^2)(10) = \frac{1}{3} \pi (36)(10) = \frac{360\pi}{3} = 120\pi \, \text{cm}^3 \]

### Step 3: Calculate the volume of the remaining solid
To find the volume of the remaining solid after the cone is removed from the cylinder, we subtract the volume of the cone from the volume of the cylinder:

\[ V_{remaining} = V_{cylinder} - V_{cone} \]

Substituting the volumes we calculated:

\[ V_{remaining} = 360\pi - 120\pi = 240\pi \, \text{cm}^3 \]

### Final Answer
The volume of the remaining solid is:

\[ \boxed{240\pi \, \text{cm}^3} \]