The dimensions of a rectangular box are in the ratio `4: 2: 3`. The difference between cost of covering it with paper at Rs 12 per `m^(2)` and with paper at the rate of 13.50 per `m^(2)` is Rs 1,248. Find the dimensions of the box.

Answer

To solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Define the dimensions of the box
Let the dimensions of the rectangular box be in the ratio of 4:2:3. We can express the dimensions as:
- Length = 4x
- Width = 2x
- Height = 3x

### Step 2: Calculate the surface area of the box
The formula for the surface area (SA) of a rectangular box is:
\[ SA = 2(lw + lh + wh) \]
Substituting the dimensions into the formula:
\[ SA = 2((4x)(2x) + (4x)(3x) + (2x)(3x)) \]
Calculating each term:
- \(lw = (4x)(2x) = 8x^2\)
- \(lh = (4x)(3x) = 12x^2\)
- \(wh = (2x)(3x) = 6x^2\)

Now, substituting back into the surface area formula:
\[ SA = 2(8x^2 + 12x^2 + 6x^2) = 2(26x^2) = 52x^2 \]

### Step 3: Set up the cost equations
The cost of covering the box with paper at Rs 12 per m² and Rs 13.50 per m² leads to the following costs:
- Cost at Rs 12 per m²: \(12 \times SA = 12 \times 52x^2 = 624x^2\)
- Cost at Rs 13.50 per m²: \(13.50 \times SA = 13.50 \times 52x^2 = 702x^2\)

### Step 4: Find the difference in costs
According to the problem, the difference between these two costs is Rs 1248:
\[ 702x^2 - 624x^2 = 1248 \]
Simplifying this gives:
\[ 78x^2 = 1248 \]

### Step 5: Solve for x
To find the value of x, divide both sides by 78:
\[ x^2 = \frac{1248}{78} = 16 \]
Taking the square root:
\[ x = 4 \]

### Step 6: Calculate the dimensions of the box
Now that we have the value of x, we can find the dimensions:
- Length = \(4x = 4 \times 4 = 16\)
- Width = \(2x = 2 \times 4 = 8\)
- Height = \(3x = 3 \times 4 = 12\)

Thus, the dimensions of the box are:
- Length = 16 m
- Width = 8 m
- Height = 12 m

### Final Answer
The dimensions of the box are 16 m, 8 m, and 12 m.