42 cubes each of side 1 cm area glured together to form a solid cuboid . If the perimeter of the base of the cuboid is 18 cm, then it height (in cm) is

Options

A. 3
B. 4
C. 1
D. 2

Answer

To solve the problem, we need to determine the height of a cuboid formed by gluing together 42 cubes, each with a side length of 1 cm. We are also given that the perimeter of the base of the cuboid is 18 cm.

### Step-by-Step Solution:

1. **Calculate the Volume of the Cuboid:**
Each cube has a volume of \(1 \, \text{cm}^3\) (since volume = side³). Therefore, the total volume of 42 cubes is:
\[ \text{Total Volume} = 42 \times 1^3 = 42 \, \text{cm}^3 \]

2. **Understand the Perimeter of the Base:**
The perimeter \(P\) of a rectangle (base of the cuboid) is given by the formula:
\[ P = 2 \times (\text{length} + \text{width}) \]
Given that the perimeter is 18 cm, we can set up the equation:
\[ 2 \times (l + w) = 18 \]
Simplifying this gives:
\[ l + w = 9 \quad \text{(Equation 1)} \]

3. **Express the Volume of the Cuboid:**
The volume \(V\) of the cuboid can also be expressed as:
\[ V = l \times w \times h \]
We know the volume is 42 cm³, so:
\[ l \times w \times h = 42 \quad \text{(Equation 2)} \]

4. **Substituting for Height:**
From Equation 1, we can express \(w\) in terms of \(l\):
\[ w = 9 - l \]
Substituting this into Equation 2 gives:
\[ l \times (9 - l) \times h = 42 \]
This simplifies to:
\[ (9l - l^2) \times h = 42 \]

5. **Finding Possible Dimensions:**
To find suitable dimensions, we can try different integer values for \(l\) and \(w\) such that \(l + w = 9\) and both \(l\) and \(w\) are integers. The pairs are:
- \(l = 1, w = 8\)
- \(l = 2, w = 7\)
- \(l = 3, w = 6\)
- \(l = 4, w = 5\)

6. **Calculating Height for Each Pair:**
For each pair, we can calculate \(h\):
- For \(l = 1, w = 8\):
\[ 1 \times 8 \times h = 42 \implies h = \frac{42}{8} = 5.25 \, \text{(not an integer)} \]
- For \(l = 2, w = 7\):
\[ 2 \times 7 \times h = 42 \implies h = \frac{42}{14} = 3 \, \text{(valid)} \]
- For \(l = 3, w = 6\):
\[ 3 \times 6 \times h = 42 \implies h = \frac{42}{18} = 2.33 \, \text{(not an integer)} \]
- For \(l = 4, w = 5\):
\[ 4 \times 5 \times h = 42 \implies h = \frac{42}{20} = 2.1 \, \text{(not an integer)} \]

7. **Conclusion:**
The only valid integer height is when \(l = 2\) and \(w = 7\), giving us:
\[ h = 3 \, \text{cm} \]

### Final Answer:
The height of the cuboid is \(3 \, \text{cm}\).