A cuboid has edges of x cm, 1 cm and 2 cm. The total surface area of the cuboid has a numerical value which is some integral multiple of the numerical value of its volume. What is the value of x for minimum possible positive integral multiple ?

Options

A. 5 cm
B. 2 cm
C. 3 cm
D. 4 cm

Answer

To solve the problem step by step, we will calculate the total surface area and volume of the cuboid, and then find the value of \( x \) for the minimum possible positive integral multiple.

### Step 1: Understand the dimensions of the cuboid
The dimensions of the cuboid are given as:
- Length \( l = x \) cm
- Breadth \( b = 1 \) cm
- Height \( h = 2 \) cm

### Step 2: Calculate the Total Surface Area (TSA)
The formula for the total surface area of a cuboid is:
\[ \text{TSA} = 2(lb + bh + hl) \]
Substituting the values:
\[ \text{TSA} = 2(x \cdot 1 + 1 \cdot 2 + 2 \cdot x) \]
\[ = 2(x + 2 + 2x) \]
\[ = 2(3x + 2) \]
\[ = 6x + 4 \text{ cm}^2 \]

### Step 3: Calculate the Volume (V)
The formula for the volume of a cuboid is:
\[ V = l \cdot b \cdot h \]
Substituting the values:
\[ V = x \cdot 1 \cdot 2 = 2x \text{ cm}^3 \]

### Step 4: Set up the equation for the integral multiple
According to the problem, the total surface area is an integral multiple of the volume:
\[ 6x + 4 = n \cdot (2x) \]
where \( n \) is a positive integer.

### Step 5: Rearranging the equation
Rearranging the equation gives:
\[ 6x + 4 = 2nx \]
\[ 6x - 2nx + 4 = 0 \]
\[ (6 - 2n)x + 4 = 0 \]

### Step 6: Solve for \( x \)
From the equation:
\[ (6 - 2n)x = -4 \]
This implies:
\[ x = \frac{-4}{6 - 2n} \]
Since \( x \) must be positive, we need \( 6 - 2n < 0 \):
\[ 6 < 2n \implies n > 3 \]

### Step 7: Find the minimum value of \( n \)
The minimum integer value for \( n \) that satisfies \( n > 3 \) is \( n = 4 \).

### Step 8: Substitute \( n = 4 \) back to find \( x \)
Substituting \( n = 4 \) into the equation:
\[ x = \frac{-4}{6 - 2 \cdot 4} = \frac{-4}{6 - 8} = \frac{-4}{-2} = 2 \text{ cm} \]

### Conclusion
The value of \( x \) for the minimum possible positive integral multiple is \( 2 \) cm.

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