The surface areas of three faces of a cuboid that meet at a common vertex are 20 sq. m, 32 sq. m and 40 sq. m respectively. What is the volume of the cuboid in cubic metres?

Introduction / Context:
This problem deals with a cuboid whose three pairwise perpendicular faces have known areas. The faces share a common vertex, so their areas correspond to products of the three edge lengths. The question tests your ability to use these face areas to deduce the edge lengths and thereby find the volume of the cuboid. It is a classic aptitude question on three dimensional geometry.

Given Data / Assumptions:

- Let the three edges meeting at a vertex be of lengths a, b and c metres.
- Areas of the three faces at that vertex are: ab = 20 sq. m, bc = 32 sq. m and ca = 40 sq. m.
- All angles between the edges are right angles as in a standard cuboid.
- We need to find the volume of the cuboid in cubic metres, which is a * b * c.

Concept / Approach:
Given ab, bc and ca, we want abc. Notice that (ab) * (bc) * (ca) = a^2 * b^2 * c^2 = (abc)^2. So, by multiplying the three given face areas, we obtain (abc)^2, and taking the positive square root gives abc, the volume. This approach avoids finding each individual edge separately and is both elegant and efficient.

Step-by-Step Solution:
Step 1: Let ab = 20, bc = 32 and ca = 40 (all in sq. m).Step 2: Multiply the three expressions: (ab) * (bc) * (ca) = 20 * 32 * 40.Step 3: Left side becomes a^2 * b^2 * c^2 = (abc)^2.Step 4: Compute the product on the right: 20 * 40 = 800, and 800 * 32 = 25600.Step 5: Thus (abc)^2 = 25600.Step 6: Take the positive square root: abc = square root of 25600.Step 7: 25600 = 160^2, so abc = 160 cubic metres.

Verification / Alternative check:
We can individually recover one set of edge lengths. From ab = 20 and ca = 40, divide to get (ca) / (ab) = c / b = 40 / 20 = 2. Thus c = 2b. From bc = 32, substitute to get b * 2b = 32 so 2b^2 = 32 and b^2 = 16, giving b = 4 m. Then c = 8 m and a = 20 / b = 5 m. Volume = a * b * c = 5 * 4 * 8 = 160 cubic metres, confirming the earlier method.

Why Other Options Are Wrong:
120, 140, 184 and 200 cubic metres do not satisfy the relationships implied by the three face areas. If you pick any of those volumes and attempt to factor them into edges that reproduce ab = 20, bc = 32 and ca = 40, you will fail. These values arise mainly from arithmetic mistakes in the multiplication or in extracting the square root.

Common Pitfalls:
Learners sometimes try to guess side lengths directly without using the elegant (abc)^2 relationship, which can lead to confusion. Another frequent mistake is miscalculating the product 20 * 32 * 40 or taking an incorrect square root of 25600. Ensuring careful multiplication and recognising 160^2 prevents errors. Remember that in physical geometry we only take the positive square root because edge lengths and volumes are positive quantities.

Final Answer:
Therefore, the volume of the cuboid is 160 cubic metres.