The length breadth and height of a cuboid are in the ratio 6: 5: 3 If its total surface area is ` 504 cm ^(2) ` , find its dimmension . Also find the volume of the cuboid.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the ratio of dimensions
The length, breadth, and height of the cuboid are given in the ratio 6:5:3. We can express these dimensions in terms of a variable \( x \):
- Length \( L = 6x \)
- Breadth \( B = 5x \)
- Height \( H = 3x \)

### Step 2: Use the formula for total surface area
The total surface area (TSA) of a cuboid is given by the formula:
\[ TSA = 2(LB + BH + LH) \]
We know the total surface area is \( 504 \, \text{cm}^2 \). Substituting the expressions for \( L \), \( B \), and \( H \):
\[ 504 = 2( (6x)(5x) + (5x)(3x) + (6x)(3x) ) \]

### Step 3: Simplify the equation
Calculating each term inside the parentheses:
- \( (6x)(5x) = 30x^2 \)
- \( (5x)(3x) = 15x^2 \)
- \( (6x)(3x) = 18x^2 \)

Now, substituting these back into the equation:
\[ 504 = 2(30x^2 + 15x^2 + 18x^2) \]
Combine the terms:
\[ 504 = 2(63x^2) \]
This simplifies to:
\[ 504 = 126x^2 \]

### Step 4: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 126:
\[ x^2 = \frac{504}{126} \]
Calculating the right side:
\[ x^2 = 4 \]

### Step 5: Find \( x \)
Taking the square root of both sides:
\[ x = 2 \]

### Step 6: Calculate the dimensions
Now we can find the dimensions of the cuboid:
- Length \( L = 6x = 6 \times 2 = 12 \, \text{cm} \)
- Breadth \( B = 5x = 5 \times 2 = 10 \, \text{cm} \)
- Height \( H = 3x = 3 \times 2 = 6 \, \text{cm} \)

### Step 7: Calculate the volume
The volume \( V \) of the cuboid is given by the formula:
\[ V = L \times B \times H \]
Substituting the values we found:
\[ V = 12 \times 10 \times 6 \]
Calculating this gives:
\[ V = 720 \, \text{cm}^3 \]

### Final Answer
The dimensions of the cuboid are:
- Length: \( 12 \, \text{cm} \)
- Breadth: \( 10 \, \text{cm} \)
- Height: \( 6 \, \text{cm} \)

The volume of the cuboid is:
- Volume: \( 720 \, \text{cm}^3 \)

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