Cube of side 14 cm with two maximal opposite hemispherical cavities:\nTwo identical hemispheres of the largest possible size are removed from opposite faces of a solid cube of side 14 cm. Find the total volume (in cm^3) of the remaining solid.

Introduction / Context:
Removing two hemispheres of maximal size from opposite faces is equivalent to removing one full sphere whose diameter equals the cube’s side. Compute cube volume, subtract the sphere volume, and simplify.

Given Data / Assumptions:

• Cube side a = 14 cm ⇒ Volume(cube) = a^3 = 2744 cm^3.
• Maximal hemisphere radius r = a/2 = 7 cm.
• Two hemispheres = one sphere of radius 7 cm.
• Use π = 22/7.

Concept / Approach:
Volume remaining = a^3 − (4/3)πr^3.

Step-by-Step Solution:

Verification / Alternative check:
Quick check with π ≈ 3.14 yields a similar result (~1306.6), confirming consistency.

Why Other Options Are Wrong:
1556.33, 1467.33, 898.5 do not equal 2744 − (4/3)π 7^3 with π = 22/7.

Common Pitfalls:
Subtracting two hemispheres separately and rounding early, or using diameter instead of radius in the sphere formula.

Final Answer:
1306.67