Assertion : The sum of the length, breadth and height of a cuboid is 19 cm and its diagonal is `5sqrt5` cm. Its surface area is 236 `cm^(2).
Reason : The lateral surface area of a cuboid is 2(l + b).

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If assertion is false but reason is true.

Answer

To solve the problem step by step, we will analyze the assertions and reasons given in the question.

### Step 1: Understand the Given Information
We are given:
1. The sum of the length (l), breadth (b), and height (h) of a cuboid is 19 cm.
\[ l + b + h = 19 \quad \text{(Equation 1)} \]
2. The diagonal (d) of the cuboid is \(5\sqrt{5}\) cm.
\[ d = \sqrt{l^2 + b^2 + h^2} = 5\sqrt{5} \quad \text{(Equation 2)} \]
3. The surface area (SA) of the cuboid is 236 cm².
\[ SA = 2(lb + bh + hl) = 236 \quad \text{(Equation 3)} \]

### Step 2: Square the Diagonal Equation
From Equation 2, we square both sides:
\[ l^2 + b^2 + h^2 = (5\sqrt{5})^2 = 125 \quad \text{(Equation 4)} \]

### Step 3: Square Equation 1
Now, we square Equation 1:
\[ (l + b + h)^2 = 19^2 = 361 \]
Using the identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\), we can expand this:
\[ l^2 + b^2 + h^2 + 2(lb + bh + hl) = 361 \]

### Step 4: Substitute Equation 4 into the Expanded Equation
Now, substitute Equation 4 into the expanded equation:
\[ 125 + 2(lb + bh + hl) = 361 \]

### Step 5: Solve for \(2(lb + bh + hl)\)
Rearranging gives:
\[ 2(lb + bh + hl) = 361 - 125 = 236 \]

### Step 6: Find the Surface Area
Since we have \(2(lb + bh + hl) = 236\), we can conclude that the surface area is indeed:
\[ SA = 236 \text{ cm}^2 \]
Thus, Assertion 1 is true.

### Step 7: Analyze the Reason
The reason states that the lateral surface area of a cuboid is \(2(l + b)\). However, the correct formula for the lateral surface area is:
\[ Lateral \, Surface \, Area = 2h(l + b) \]
Therefore, the reason is false.

### Conclusion
- Assertion: True
- Reason: False

### Final Answer
Assertion 1 is true, while Reason is false.

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