There is a solid cube with side 10 m. If the largest possible cone is carved out of it, then what is the surface area of the remaining part of the cube?

Options

A. `600 + 25sqrt(5pi)`
B. `500 + 25 sqrt(5pi)`
C. `600 - 25 (sqrt(5+1)pi`
D. `600 + 25 (sqrt(5)-1)pi`

Answer

To solve the problem of finding the surface area of the remaining part of the cube after carving out the largest possible cone, we can follow these steps:

### Step 1: Calculate the Total Surface Area (TSA) of the Cube
The formula for the total surface area of a cube is given by:
\[ \text{TSA} = 6a^2 \]
where \( a \) is the length of a side of the cube.

Given that the side of the cube is \( 10 \, \text{m} \):
\[ \text{TSA} = 6 \times (10)^2 = 6 \times 100 = 600 \, \text{m}^2 \]

### Step 2: Determine the Dimensions of the Cone
The largest cone that can be carved out of the cube will have:
- Height \( h = 10 \, \text{m} \) (equal to the side of the cube)
- Radius \( r = \frac{10}{2} = 5 \, \text{m} \) (half of the side of the cube)

### Step 3: Calculate the Slant Height (L) of the Cone
The slant height \( L \) of the cone can be calculated using the Pythagorean theorem:
\[ L = \sqrt{h^2 + r^2} \]
Substituting the values:
\[ L = \sqrt{(10)^2 + (5)^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \, \text{m} \]

### Step 4: Calculate the Curved Surface Area (CSA) of the Cone
The formula for the curved surface area of a cone is given by:
\[ \text{CSA} = \pi r L \]
Substituting the values:
\[ \text{CSA} = \pi \times 5 \times 5\sqrt{5} = 25\pi\sqrt{5} \, \text{m}^2 \]

### Step 5: Calculate the Surface Area of the Remaining Part of the Cube
To find the surface area of the remaining part of the cube after the cone is carved out, we subtract the curved surface area of the cone from the total surface area of the cube. However, we need to add the base area of the cone back because it is not part of the cube's surface area.

The base area of the cone is:
\[ \text{Base Area} = \pi r^2 = \pi \times (5)^2 = 25\pi \, \text{m}^2 \]

Now, the surface area of the remaining part of the cube is:
\[ \text{Remaining Surface Area} = \text{TSA of Cube} - \text{CSA of Cone} + \text{Base Area of Cone} \]
Substituting the values:
\[ \text{Remaining Surface Area} = 600 - 25\pi\sqrt{5} + 25\pi \]

### Final Expression
Thus, the surface area of the remaining part of the cube is:
\[ \text{Remaining Surface Area} = 600 - 25\pi\sqrt{5} + 25\pi \, \text{m}^2 \]