From a solid cylinder, whose height is 8 cm and radius is 6 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume of the remaining solid.

Answer

To find the volume of the remaining solid after hollowing out a conical cavity from a solid cylinder, we can follow these steps:

### Step 1: Calculate the Volume of the Solid Cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder

Given:
- Radius \( r = 6 \) cm
- Height \( h = 8 \) cm

Substituting the values into the formula:
\[ V_{\text{cylinder}} = \pi (6)^2 (8) = \pi (36)(8) = 288\pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of the Conical Cavity
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the cone
- \( h \) is the height of the cone

Given:
- Radius \( r = 6 \) cm
- Height \( h = 8 \) cm

Substituting the values into the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi (6)^2 (8) = \frac{1}{3} \pi (36)(8) = \frac{288\pi}{3} = 96\pi \, \text{cm}^3 \]

### Step 3: Calculate the Volume of the Remaining Solid
Now, we can find the volume of the remaining solid by subtracting the volume of the conical cavity from the volume of the solid cylinder:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} \]
Substituting the volumes we calculated:
\[ V_{\text{remaining}} = 288\pi - 96\pi = 192\pi \, \text{cm}^3 \]

### Step 4: Approximate the Volume
To get a numerical approximation, we can use \( \pi \approx \frac{22}{7} \):
\[ V_{\text{remaining}} \approx 192 \times \frac{22}{7} = \frac{4224}{7} \approx 603.43 \, \text{cm}^3 \]

### Final Answer
The volume of the remaining solid is approximately \( 603.43 \, \text{cm}^3 \).

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