The edges of a rectangular box are in the ratio `1:2:3` and area of its floor is `8 cm^(2)`. The volume of the box is

Options

A. `24 cm^(3)`
B. `48 cm^(3)`
C. `64 cm^(3)`
D. `120 cm^(3)`

Answer

To find the volume of a rectangular box with edges in the ratio 1:2:3 and a floor area of 8 cm², we can follow these steps:

### Step-by-Step Solution:

1. **Define the Dimensions**:
Let the dimensions of the rectangular box be represented as:
- Length (L) = 1x
- Breadth (B) = 2x
- Height (H) = 3x
Here, x is a common multiplier.

2. **Calculate the Area of the Floor**:
The area of the floor of the box (which is a rectangle) is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
Substituting the values, we have:
\[ \text{Area} = L \times B = (1x) \times (2x) = 2x^2 \]
We know from the problem that the area of the floor is 8 cm², so we can set up the equation:
\[ 2x^2 = 8 \]

3. **Solve for x**:
To find x, we can rearrange the equation:
\[ x^2 = \frac{8}{2} = 4 \]
Taking the square root of both sides gives:
\[ x = 2 \]

4. **Find the Dimensions**:
Now that we have the value of x, we can find the actual dimensions:
- Length \( L = 1x = 1 \times 2 = 2 \, \text{cm} \)
- Breadth \( B = 2x = 2 \times 2 = 4 \, \text{cm} \)
- Height \( H = 3x = 3 \times 2 = 6 \, \text{cm} \)

5. **Calculate the Volume**:
The volume \( V \) of the box is given by the formula:
\[ V = L \times B \times H \]
Substituting the values we found:
\[ V = 2 \times 4 \times 6 \]
Performing the multiplication:
\[ V = 48 \, \text{cm}^3 \]

### Final Answer:
The volume of the box is \( 48 \, \text{cm}^3 \).

---