The sum of the length, breadth and height of a cuboid is 22 cm, and the length of its space diagonal is 14 cm. What is the total surface area (in square cm) of the cuboid?

Introduction / Context:
This problem uses algebraic identities related to a cuboid. You are given the sum of its three dimensions and the length of its space diagonal. From this, you must deduce the total surface area. It checks your ability to connect geometric formulas with algebraic identities.

Given Data / Assumptions:

• Let the length, breadth and height of the cuboid be l, b and h (in cm).
• l + b + h = 22 cm.
• The space diagonal d satisfies d^2 = l^2 + b^2 + h^2 and we are told d = 14 cm.
• So l^2 + b^2 + h^2 = 14^2 = 196.
• We assume a standard right angled cuboid (rectangular box).
• We want the total surface area S = 2(lb + bh + hl).

Concept / Approach:
The key algebraic identity is: (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) We already know the left side and l^2 + b^2 + h^2, so we can solve for lb + bh + hl. Once that sum is known, we multiply by 2 to get the surface area of the cuboid.

Step-by-Step Solution:
Step 1: Square the sum of dimensions: (l + b + h)^2 = 22^2 = 484. Step 2: Use the identity (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl). Step 3: Substitute l^2 + b^2 + h^2 = 196 into the identity. Step 4: So 484 = 196 + 2(lb + bh + hl). Step 5: Rearrange to find 2(lb + bh + hl) = 484 − 196 = 288. Step 6: The total surface area S of the cuboid is S = 2(lb + bh + hl) = 288 sq cm.

Verification / Alternative check:
Note that we did not need individual values of l, b and h. The combination of sum of squares and sum of dimensions is sufficient to determine 2(lb + bh + hl) uniquely. This shows that the surface area is fixed by the given data. No contradiction appears in the calculations, and all steps smoothly follow from standard identities, so 288 sq cm is reliable.

Why Other Options Are Wrong:
216 sq cm and 144 sq cm: These would correspond to smaller values of lb + bh + hl, which contradict the computed value 144 (since 2 times that gives 288).
Cannot be determined from the given information: This is incorrect because the identity involving the sum of dimensions and sum of squares is enough to fix the surface area uniquely.
200 sq cm: This is not supported by any consistent algebraic manipulation of the given data.

Common Pitfalls:
Students sometimes try to assume specific integer values for l, b and h, which is unnecessary and may lead to trial and error. Another error is forgetting the algebraic identity and thinking that surface area cannot be determined. Miscomputing 22^2 or 14^2, or making sign errors while rearranging, can also produce wrong answers. Keeping the identity clearly in mind avoids these mistakes.

Final Answer:
The total surface area of the cuboid is 288 sq cm.