Question

Assertion (A) : The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end-to-end, is 160 \(\text{cm}^2\).
Reason (R) : The surface area of a cuboid of dimensions \(l \times b \times h\) is \((lb + bh + hl)\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

An alternative way to calculate the surface area of the joined cubes:
Two separate cubes have \(6 + 6 = 12\) faces in total.
When joined together, 2 faces (one from each cube) overlap and are hidden inside.
So, the surface area is the area of the remaining 10 exposed faces:
\[ \text{Surface Area} = 10 \times s^2 = 10 \times 4^2 = 10 \times 16 = 160\ \text{cm}^2 \]
This structural logic is very quick and visual!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given an Assertion-Reason format question. We need to evaluate the truth of Assertion (A) and Reason (R) independently, and then check their relationship.

Step 2: Key Formula or Approach:
1. The surface area of a cuboid with length \(l\), width \(b\), and height \(h\) is:
\[ \text{Total Surface Area } (\text{TSA}) = 2(lb + bh + hl) \]
2. When two identical cubes of side \(s\) are joined end-to-end, they form a cuboid of:
- Length, \(l = 2s\)
- Breadth, \(b = s\)
- Height, \(h = s\)

Step 3: Detailed Explanation:
1.

Evaluate Assertion (A):
Two identical cubes of side \(s = 4\ \text{cm}\) are joined end-to-end.
The dimensions of the resulting cuboid are:
- Length, \(l = 4 + 4 = 8\ \text{cm}\)
- Breadth, \(b = 4\ \text{cm}\)
- Height, \(h = 4\ \text{cm}\)
Now, calculate its surface area:
\[ \text{Surface Area} = 2(lb + bh + hl) \]
\[ \text{Surface Area} = 2(8 \times 4 + 4 \times 4 + 4 \times 8) \]
\[ \text{Surface Area} = 2(32 + 16 + 32) \]
\[ \text{Surface Area} = 2(80) = 160\ \text{cm}^2 \]
Since this matches the assertion statement, Assertion (A) is true.

2.

Evaluate Reason (R):
The reason states: "The surface area of a cuboid of dimensions \(l \times b \times h\) is \((lb + bh + hl)\)."
This formula is mathematically incorrect because it is missing the essential factor of 2.
The correct surface area formula is \(2(lb + bh + hl)\).
Therefore, Reason (R) is false.

3. Combining our findings: Assertion (A) is true, but Reason (R) is false. This leads to option (C).

Step 4: Final Answer:
Assertion (A) is true, but Reason (R) is false, which corresponds to option (C).