From a solid cylinder whose height is 16 cm and radius is 12 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume and total surface area of the remaining solid.

Answer

To find the volume and total surface area 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 Cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Where:
- \( r = 12 \, \text{cm} \) (radius of the cylinder)
- \( h = 16 \, \text{cm} \) (height of the cylinder)

Substituting the values:
\[ V_{\text{cylinder}} = \pi (12)^2 (16) = \pi (144)(16) = 2304\pi \, \text{cm}^3 \]

### 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:
- \( r = 6 \, \text{cm} \) (radius of the cone)
- \( h = 8 \, \text{cm} \) (height of the cone)

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

### Step 3: Calculate the Volume of the Remaining Solid
The volume of the remaining solid is the volume of the cylinder minus the volume of the conical cavity:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 2304\pi - 96\pi = (2304 - 96)\pi = 2208\pi \, \text{cm}^3 \]

### Step 4: Calculate the Slant Height of the Cone
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \, \text{cm} \]

### Step 5: Calculate the Total Surface Area of the Remaining Solid
The total surface area of the remaining solid can be calculated as:
\[ \text{TSA} = \text{TSA of Cylinder} - \text{Base Area of Cone} + \text{CSA of Cone} \]
Where:
- TSA of Cylinder = \( 2\pi r (r + h) \)
- Base Area of Cone = \( \pi r^2 \)
- CSA of Cone = \( \pi r l \)

Calculating each part:
1. **TSA of Cylinder**:
\[ \text{TSA}_{\text{cylinder}} = 2\pi (12)(12 + 16) = 2\pi (12)(28) = 672\pi \, \text{cm}^2 \]

2. **Base Area of Cone**:
\[ \text{Base Area}_{\text{cone}} = \pi (6)^2 = 36\pi \, \text{cm}^2 \]

3. **CSA of Cone**:
\[ \text{CSA}_{\text{cone}} = \pi (6)(10) = 60\pi \, \text{cm}^2 \]

Now substituting these values into the TSA formula:
\[ \text{TSA} = 672\pi - 36\pi + 60\pi = (672 - 36 + 60)\pi = 696\pi \, \text{cm}^2 \]

### Final Results
- Volume of the remaining solid: \( 2208\pi \, \text{cm}^3 \) (approximately \( 6934.3 \, \text{cm}^3 \))
- Total Surface Area of the remaining solid: \( 696\pi \, \text{cm}^2 \) (approximately \( 2186.8 \, \text{cm}^2 \))