A cuboid has volume 720 cm^3, total surface area 484 cm^2, and base area 72 cm^2. Find its dimensions (length, breadth, height) in cm.

Introduction / Context:
Given base area lb, volume V, and total surface area S, a cuboid’s three dimensions can be recovered systematically. From V = lb * h we get height; then S = 2(lb + lh + bh) yields l + b. With l + b and lb known, l and b are roots of a quadratic. This integrates multiple cuboid identities coherently.

Given Data / Assumptions:

• lb = 72 cm^2.
• V = 720 cm^3 ⇒ h = V / lb = 10 cm.
• S = 2(lb + lh + bh) = 484 cm^2.

Concept / Approach:
Use h = 10 in S to solve for l + b. Then solve t^2 − (l + b)t + lb = 0 for t, whose roots are l and b. Pick the positive pair producing the given base area.

Step-by-Step Solution:
484 = 2(72 + 10l + 10b)Divide by 2: 242 = 72 + 10(l + b) ⇒ 10(l + b) = 170 ⇒ l + b = 17Solve t^2 − 17t + 72 = 0 ⇒ (t − 9)(t − 8) = 0Thus (l, b) = (9, 8) in some order; with h = 10

Verification / Alternative check:
Check: lb = 72, V = 72 * 10 = 720, S = 2(72 + 90 + 80) = 2 * 242 = 484, all consistent.

Why Other Options Are Wrong:
Other triples give incorrect base area or surface computations; only 9, 8, 10 fits all three conditions.

Common Pitfalls:
Forgetting to halve S; forming the wrong quadratic; mixing the roles of base and height.

Final Answer:
9, 8 and 10 cm