The surface area of the cuboid is 1372 sq. cm. if its dimension are in the ratio of 4:2:1. then find its length.

Answer

To find the length of the cuboid given its surface area and the ratio of its dimensions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio of Dimensions**:
The dimensions of the cuboid are given in the ratio 4:2:1. We can denote:
- Length (L) = 4x
- Breadth (B) = 2x
- Height (H) = x

2. **Write the Formula for Surface Area**:
The formula for the surface area (SA) of a cuboid is:
\[ SA = 2(LB + BH + LH) \]
We know the surface area is 1372 sq. cm, so we can set up the equation:
\[ 2(LB + BH + LH) = 1372 \]

3. **Substitute the Dimensions into the Surface Area Formula**:
Substitute L, B, and H with their expressions in terms of x:
\[ 2((4x)(2x) + (2x)(x) + (x)(4x)) = 1372 \]

4. **Simplify the Equation**:
Calculate each term inside the parentheses:
- \( (4x)(2x) = 8x^2 \)
- \( (2x)(x) = 2x^2 \)
- \( (x)(4x) = 4x^2 \)

Combine these:
\[ 8x^2 + 2x^2 + 4x^2 = 14x^2 \]
Now, substitute back into the surface area equation:
\[ 2(14x^2) = 1372 \]

5. **Solve for x**:
Divide both sides by 2:
\[ 14x^2 = 686 \]
Now, divide by 14:
\[ x^2 = \frac{686}{14} = 49 \]
Taking the square root:
\[ x = 7 \text{ cm} \]

6. **Find the Length**:
Now, substitute x back to find the length:
\[ L = 4x = 4 \times 7 = 28 \text{ cm} \]

### Final Answer:
The length of the cuboid is **28 cm**.