Two identical solid hemispheres of the maximum possible radius are cut out from a solid cube whose edge is 14 cm. The flat circular bases of the hemispheres lie on the centres of two opposite faces of the cube. What is the total volume, in cubic centimetres (cm³), of the remaining solid part of the cube?

Introduction / Context:
This question tests your understanding of volumes of basic three dimensional solids, specifically a cube and a hemisphere, and how to find the volume of a remaining solid when certain parts are removed. Such problems are common in aptitude and engineering entrance exams because they combine geometry, visualization and careful calculation of volumes.

Given Data / Assumptions:

• Side of the solid cube = 14 cm.

• Two identical hemispheres are cut from two opposite faces of the cube.

• The hemispheres are of maximum possible size, so their diameter equals the side of the cube.

• Radius of each hemisphere r = 7 cm.

• Take pi = 22/7 for numerical evaluation.

Concept / Approach:
The key idea is to compute the original volume of the cube and then subtract the total volume removed in the form of two hemispheres. The volume of a cube is side^3. The volume of a full sphere is (4/3) * pi * r^3, hence the volume of one hemisphere is half of that, that is (2/3) * pi * r^3. Since there are two identical hemispheres, their combined volume equals the volume of one complete sphere of radius 7 cm.

Step-by-Step Solution:
Volume of cube = side^3 = 14^3 cm³ = 2744 cm³. Radius of each hemisphere r = 7 cm. Volume of one hemisphere = (2/3) * pi * r^3. Volume of one hemisphere = (2/3) * pi * 7^3 = (2/3) * pi * 343. Total volume removed (two hemispheres) = 2 * (2/3) * pi * 343 = (4/3) * pi * 343. Using pi = 22/7, total removed volume = (4/3) * (22/7) * 343. Simplify 343 / 7 = 49, so removed volume = (4/3) * 22 * 49. Removed volume = (4 * 22 * 49) / 3 = 4312 / 3 ≈ 1437.33 cm³. Remaining volume = volume of cube − volume removed. Remaining volume = 2744 − 1437.33 ≈ 1306.67 cm³.

Verification / Alternative check:
An alternative way is to notice that two hemispheres together form a complete sphere of radius 7 cm. So you could directly compute the volume of a sphere of radius 7 cm as (4/3) * pi * 7^3 and subtract it from the cube volume. This is the same expression as derived above, which confirms that the calculation is consistent. Rechecking the arithmetic with pi = 22/7 again gives approximately 1437.33 cm³ removed and 1306.67 cm³ left.

Why Other Options Are Wrong:
The value 1556.33 cm³ is larger than the volume removed and does not match the subtraction from the cube volume. The value 898.50 cm³ is too small for the remaining volume and arises from incorrect arithmetic or perhaps using radius or diameter wrongly. The value 1467.33 cm³ is closer but still incorrect and would correspond to an inaccurate approximation or using the wrong initial volume. Only 1306.67 cm³ matches the correct step by step computation.

Common Pitfalls:
Students often confuse diameter with radius and may use 14 cm as the radius of the hemisphere, which doubles the actual radius and leads to a huge overestimation of removed volume. Another frequent error is to forget that two hemispheres together make one full sphere, and they may subtract only the volume of a single hemisphere. Using an incorrect value of pi or rounding intermediate steps too early can also distort the final answer. Always check units and ensure that all dimensions are in centimetres before applying formulas.

Final Answer:
The total volume of the remaining solid after cutting out the two hemispheres from the cube is 1306.67 cm³ (approximately).