The length, breadth and height of a wooden box with a lid are 10 cm, 9 cm and 7 cm, respectively. The total inner surface of the closed box is `262 cm^(2)`. The thickness of the wood (in cm.) is

Options

A. 2
B. 3
C. `(23)/(3)`
D. 1

Answer

To find the thickness of the wood in the wooden box, we will follow these steps:

### Step 1: Define the Variables
Let the thickness of the wood be \( x \) cm.

### Step 2: Calculate Inner Dimensions
The outer dimensions of the box are:
- Length (L) = 10 cm
- Breadth (B) = 9 cm
- Height (H) = 7 cm

The inner dimensions will be:
- Inner Length = \( 10 - 2x \)
- Inner Breadth = \( 9 - 2x \)
- Inner Height = \( 7 - 2x \)

### Step 3: Calculate Inner Surface Area
The formula for the total inner surface area \( A \) of a closed box is given by:
\[ A = 2(lb + bh + hl) \]
Substituting the inner dimensions:
\[ A = 2((10 - 2x)(9 - 2x) + (9 - 2x)(7 - 2x) + (7 - 2x)(10 - 2x)) \]

### Step 4: Set Up the Equation
We know the total inner surface area is 262 cm²:
\[ 2((10 - 2x)(9 - 2x) + (9 - 2x)(7 - 2x) + (7 - 2x)(10 - 2x)) = 262 \]

Dividing both sides by 2:
\[ (10 - 2x)(9 - 2x) + (9 - 2x)(7 - 2x) + (7 - 2x)(10 - 2x) = 131 \]

### Step 5: Expand the Equation
Now we will expand each term:
1. \( (10 - 2x)(9 - 2x) = 90 - 20x - 18x + 4x^2 = 90 - 38x + 4x^2 \)
2. \( (9 - 2x)(7 - 2x) = 63 - 14x - 18x + 4x^2 = 63 - 32x + 4x^2 \)
3. \( (7 - 2x)(10 - 2x) = 70 - 14x - 20x + 4x^2 = 70 - 34x + 4x^2 \)

Combining these:
\[ (90 - 38x + 4x^2) + (63 - 32x + 4x^2) + (70 - 34x + 4x^2) = 131 \]

### Step 6: Combine Like Terms
Combining all terms:
\[ 90 + 63 + 70 + (4x^2 + 4x^2 + 4x^2) - (38x + 32x + 34x) = 131 \]
\[ 223 + 12x^2 - 104x = 131 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 12x^2 - 104x + 223 - 131 = 0 \]
\[ 12x^2 - 104x + 92 = 0 \]

### Step 8: Simplify the Equation
Dividing the entire equation by 4:
\[ 3x^2 - 26x + 23 = 0 \]

### Step 9: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3, b = -26, c = 23 \):
\[ b^2 - 4ac = (-26)^2 - 4 \cdot 3 \cdot 23 = 676 - 276 = 400 \]
\[ x = \frac{26 \pm \sqrt{400}}{6} = \frac{26 \pm 20}{6} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{46}{6} = \frac{23}{3} \approx 7.67 \) (not possible since it exceeds the height)
2. \( x = \frac{6}{6} = 1 \)

### Conclusion
Thus, the thickness of the wood is \( x = 1 \) cm.