The sum of the length, breadth and height of a cuboid is 22 cm and the length of its space diagonal is 14 cm. If S is the sum of the cubes of its three dimensions and V is its volume, then what is the value of (S − 3V) in cubic cm?

Introduction / Context:
This question extends the previous cuboid problem and tests your knowledge of algebraic identities, specifically the identity for the sum of cubes in terms of the sum of numbers and their product. It mixes geometry (cuboid dimensions and diagonal) with algebra (sum of cubes and volume).

Given Data / Assumptions:

• Dimensions of the cuboid are l, b and h (in cm).
• l + b + h = 22.
• Space diagonal d = 14 cm, so l^2 + b^2 + h^2 = 14^2 = 196.
• S is defined as S = l^3 + b^3 + h^3.
• Volume V is V = l * b * h.
• We need to find S − 3V.

Concept / Approach:
There is a powerful identity for three numbers a, b and c: a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca) In our case, a, b and c are l, b and h. Here S − 3V is exactly l^3 + b^3 + h^3 − 3 l b h, so we can directly use this identity. We already know l + b + h and l^2 + b^2 + h^2. To use the identity we also need the sum of pairwise products ab + bc + ca, which can be computed from another identity: (l + b + h)^2 = l^2 + b^2 + h^2 + 2(ab + bc + ca)

Step-by-Step Solution:
Step 1: Compute (l + b + h)^2 = 22^2 = 484. Step 2: Use the identity (l + b + h)^2 = l^2 + b^2 + h^2 + 2(ab + bc + ca). Step 3: Substitute l^2 + b^2 + h^2 = 196. Step 4: 484 = 196 + 2(ab + bc + ca). Step 5: So 2(ab + bc + ca) = 484 − 196 = 288, hence ab + bc + ca = 144. Step 6: Now apply the sum of cubes identity: S − 3V = (l + b + h)(l^2 + b^2 + h^2 − ab − bc − ca). Step 7: Substitute known values: l + b + h = 22, l^2 + b^2 + h^2 = 196, and ab + bc + ca = 144. Step 8: Compute the bracket: 196 − 144 = 52. Step 9: Therefore S − 3V = 22 * 52 = 1144 cubic cm.

Verification / Alternative check:
Although we do not know the individual dimensions l, b and h, the algebraic identities guarantee that S − 3V depends only on the combinations we already computed. We can even factor 1144 as 8 * 143 to see it is a reasonable moderate sized number, not extremely large. No contradictions arise, so 1144 cubic cm is consistent.

Why Other Options Are Wrong:
572 cubic cm and 728 cubic cm: These are roughly half or smaller than the correct value and would correspond to a smaller value of the product (l + b + h)(l^2 + b^2 + h^2 − ab − bc − ca), which contradicts the exact computation.
None of the above: This is wrong because 1144 cubic cm exactly matches the algebraic result.
900 cubic cm: This is a plausible looking round number but is not supported by any correct substitution into the identities.

Common Pitfalls:
Some learners forget the correct form of the sum of cubes identity or confuse it with (a + b + c)^3. Another common issue is not computing ab + bc + ca correctly and stopping at the sum of squares. Carefully applying both identities in sequence avoids these mistakes. It is also important to remember that S − 3V is exactly the expression produced by the identity, rather than S + 3V or any other combination.

Final Answer:
The value of S − 3V is 1144 cubic cm.