Two cubes: find product of edges from given differences: The difference of volumes is 152 m^3 and the difference of one-face areas is 20 m^2. If the sum of their edges (side lengths) is 10 m, find the product of their edges.

Introduction / Context:
This problem links three algebraic relations for cube side lengths a and b (a > b): volume difference, face-area difference, and sum of edges. It tests recognition of factorization identities and system solving.

Given Data / Assumptions:

• a^3 − b^3 = 152
• a^2 − b^2 = 20 (difference of one-face areas)
• a + b = 10
• Find a * b

Concept / Approach:
Use identities: a^3 − b^3 = (a − b)(a^2 + ab + b^2); and a^2 − b^2 = (a − b)(a + b). From a^2 − b^2 and a + b, get a − b, then solve a and b. Finally compute ab.

Step-by-Step Solution:
From a^2 − b^2 = 20 and a + b = 10 → (a − b) = 20 / 10 = 2Solve a and b: a = ( (a + b) + (a − b) ) / 2 = (10 + 2)/2 = 6; b = (10 − 2)/2 = 4Check volume difference: 6^3 − 4^3 = 216 − 64 = 152 (OK)Product ab = 6 * 4 = 24

Verification / Alternative check:
Use a^2 + ab + b^2 with found a, b: 36 + 24 + 16 = 76; then (a − b)(a^2 + ab + b^2) = 2 * 76 = 152 equals the given difference, confirming a and b are correct.

Why Other Options Are Wrong:
21, 36, 48, 40 do not satisfy the three simultaneous conditions; only 24 is consistent with all constraints.

Common Pitfalls:
Misreading “sum of their edge” as sum of all 12 edges per cube; forgetting that “one-face area” is side^2; arithmetic slips solving a and b.

Final Answer:
24