A cone of maximum size is carved out from a solid cube of side 14 cm. Find the total surface area of the remaining solid left out.

Answer

To find the total surface area of the remaining solid after carving out a cone from a cube, we can follow these steps:

### Step 1: Understand the dimensions of the cube and the cone
- The side of the cube is given as 14 cm.
- The maximum size cone that can be carved out will have a height equal to the side of the cube (14 cm) and a radius equal to half the side of the cube (7 cm).

### Step 2: Calculate the dimensions of the cone
- **Height of the cone (h)** = 14 cm
- **Radius of the cone (r)** = 14 cm / 2 = 7 cm

### Step 3: Calculate the slant height of the cone
- The slant height (l) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} = \sqrt{14^2 + 7^2} = \sqrt{196 + 49} = \sqrt{245} = 7\sqrt{5} \text{ cm} \]

### Step 4: Calculate the total surface area of the cube
- The total surface area (TSA) of the cube is given by the formula:
\[ \text{TSA of cube} = 6a^2 = 6 \times (14^2) = 6 \times 196 = 1176 \text{ cm}^2 \]

### Step 5: Calculate the lateral surface area of the cone
- The lateral surface area (LSA) of the cone is given by the formula:
\[ \text{LSA of cone} = \pi r l = \frac{22}{7} \times 7 \times 7\sqrt{5} \]
Simplifying this:
\[ = 22 \times 7\sqrt{5} = 154\sqrt{5} \text{ cm}^2 \]

### Step 6: Calculate the area of the base of the cone
- The area of the base of the cone is given by:
\[ \text{Area of base of cone} = \pi r^2 = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \text{ cm}^2 \]

### Step 7: Calculate the total surface area of the remaining solid
- The total surface area of the remaining solid is given by:
\[ \text{TSA of remaining solid} = \text{TSA of cube} + \text{LSA of cone} - \text{Area of base of cone} \]
Plugging in the values:
\[ = 1176 + 154\sqrt{5} - 154 \]
\[ = 1176 - 154 + 154\sqrt{5} = 1022 + 154\sqrt{5} \text{ cm}^2 \]

### Final Answer
The total surface area of the remaining solid left out is:
\[ \text{TSA} = 1022 + 154\sqrt{5} \text{ cm}^2 \approx 1366.96 \text{ cm}^2 \]