Question

Two identical solid cubes of volume \(8\text{ cm}^3\) are joined end to end. Then the length of diagonal of the resulting cuboid is :

• A. \(3\sqrt{6}\text{ cm}\)
• B. \(2\sqrt{6}\text{ cm}\)
• C. \(2\sqrt{2}\text{ cm}\)
• D. 4 cm

Hint:

For any cuboid formed by joining \(n\) identical cubes of side \(a\) end-to-end, the diagonal is always:
\[ D = a\sqrt{n^2 + 2} \]
Here, \(a = 2\) and \(n = 2\), so:
\[ D = 2\sqrt{2^2 + 2} = 2\sqrt{6}\text{ cm} \]
This formula saves calculation time for multi-cube questions.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Mensuration, dealing with the properties of cubes and cuboids.
Two identical cubes are combined to form a single solid cuboid. We need to determine the length of the space diagonal of this new cuboid.

Step 2: Key Formula or Approach:
1. The volume of a cube with side length \(a\) is:
\[ V = a^3 \]
2. When two cubes are joined end-to-end, only one dimension (the length) doubles, while the other two dimensions (width and height) remain equal to \(a\).
3. The diagonal \(D\) of a cuboid with dimensions \(l\), \(b\), and \(h\) is:
\[ D = \sqrt{l^2 + b^2 + h^2} \]

Step 3: Detailed Explanation:
We are given that each cube has a volume of \(8\text{ cm}^3\):
\[ a^3 = 8 \implies a = 2\text{ cm} \] So, the side of each cube is \(2\text{ cm}\).
When two such cubes are joined end-to-end:
- The length of the resulting cuboid is:
\[ l = 2a = 2 \times 2 = 4\text{ cm} \]
- The breadth of the cuboid is:
\[ b = a = 2\text{ cm} \]
- The height of the cuboid is:
\[ h = a = 2\text{ cm} \]
Now, calculate the diagonal of this cuboid:
\[ D = \sqrt{l^2 + b^2 + h^2} \]
Substitute the values:
\[ D = \sqrt{4^2 + 2^2 + 2^2} \]
\[ D = \sqrt{16 + 4 + 4} \]
\[ D = \sqrt{24} \]
Simplify the radical:
\[ D = \sqrt{4 \times 6} = 2\sqrt{6}\text{ cm} \]

Step 4: Final Answer:
The length of the diagonal of the resulting cuboid is \(2\sqrt{6}\text{ cm}\).