From a solid cylinder of height 36 cm and radius 14 cm, a conical cavity of radius 7 cm and height 24 cm is drilled out. Find the volume and the total surface area of the remaining solid.

Answer

To solve the problem step by step, we will find the volume and total surface area of the remaining solid after drilling out the conical cavity from the cylinder.

### Step 1: Calculate the Volume of the Cylinder
The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi r^2 h \]
Where:
- \( r = 14 \, \text{cm} \) (radius of the cylinder)
- \( h = 36 \, \text{cm} \) (height of the cylinder)

Substituting the values:
\[ V_{cylinder} = \pi (14)^2 (36) = \pi (196)(36) = 7056\pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of the Cone
The formula for the volume of a cone is:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Where:
- \( r = 7 \, \text{cm} \) (radius of the cone)
- \( h = 24 \, \text{cm} \) (height of the cone)

Substituting the values:
\[ V_{cone} = \frac{1}{3} \pi (7)^2 (24) = \frac{1}{3} \pi (49)(24) = \frac{1176}{3}\pi = 392\pi \, \text{cm}^3 \]

### Step 3: Calculate the Volume of the Remaining Solid
The volume of the remaining solid is given by:
\[ V_{remaining} = V_{cylinder} - V_{cone} \]
Substituting the values:
\[ V_{remaining} = 7056\pi - 392\pi = (7056 - 392)\pi = 6664\pi \, \text{cm}^3 \]

Using \( \pi \approx \frac{22}{7} \):
\[ V_{remaining} \approx 6664 \times \frac{22}{7} = 20,944 \, \text{cm}^3 \]

### Step 4: Calculate the Total Surface Area of the Remaining Solid
The total surface area of the remaining solid consists of:
1. The curved surface area of the cylinder
2. The curved surface area of the cone
3. The area of the base of the cylinder (not the base of the cone)

#### Curved Surface Area of the Cylinder
\[ A_{cylinder} = 2\pi rh = 2\pi (14)(36) = 1008\pi \, \text{cm}^2 \]

#### Curved Surface Area of the Cone
To find the slant height \( l \) of the cone:
\[ l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, \text{cm} \]
Now, calculate the curved surface area of the cone:
\[ A_{cone} = \pi r l = \pi (7)(25) = 175\pi \, \text{cm}^2 \]

#### Area of the Base of the Cone
\[ A_{base\,cone} = \pi r^2 = \pi (7)^2 = 49\pi \, \text{cm}^2 \]

### Step 5: Total Surface Area of the Remaining Solid
The total surface area is given by:
\[ A_{remaining} = A_{cylinder} + A_{cone} - A_{base\,cone} \]
Substituting the values:
\[ A_{remaining} = 1008\pi + 175\pi - 49\pi = (1008 + 175 - 49)\pi = 1134\pi \, \text{cm}^2 \]

Using \( \pi \approx \frac{22}{7} \):
\[ A_{remaining} \approx 1134 \times \frac{22}{7} = 3,596.57 \, \text{cm}^2 \approx 3596 \, \text{cm}^2 \]

### Final Answers
- Volume of the remaining solid: \( 20,944 \, \text{cm}^3 \)
- Total surface area of the remaining solid: \( 3596 \, \text{cm}^2 \)