Diagonal of a cube is `6sqrt(3)cm`. Ratio of its total surface area and volume (numerically) is

Options

A. `2:1`
B. `1:6`
C. `1:1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the total surface area and volume of a cube when the diagonal is given as \(6\sqrt{3}\) cm.

### Step-by-Step Solution:

1. **Understand the relationship between the diagonal and the side of the cube:**
The formula for the diagonal \(d\) of a cube in terms of its side length \(a\) is given by:
\[ d = a\sqrt{3} \]

2. **Set up the equation with the given diagonal:**
We know the diagonal \(d\) is \(6\sqrt{3}\) cm. Therefore, we can set up the equation:
\[ a\sqrt{3} = 6\sqrt{3} \]

3. **Solve for the side length \(a\):**
To find \(a\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ a = 6 \text{ cm} \]

4. **Calculate the total surface area (TSA) of the cube:**
The formula for the total surface area of a cube is:
\[ \text{TSA} = 6a^2 \]
Substituting \(a = 6\):
\[ \text{TSA} = 6 \times (6)^2 = 6 \times 36 = 216 \text{ cm}^2 \]

5. **Calculate the volume (V) of the cube:**
The formula for the volume of a cube is:
\[ V = a^3 \]
Substituting \(a = 6\):
\[ V = (6)^3 = 216 \text{ cm}^3 \]

6. **Find the ratio of total surface area to volume:**
Now, we can find the ratio:
\[ \text{Ratio} = \frac{\text{TSA}}{V} = \frac{216}{216} = 1 \]

### Final Answer:
The ratio of the total surface area to the volume of the cube is \(1:1\).