If the areas of three adjacent faces of a cuboidal box are 120 cm², 72 cm² and 60 cm² respectively, what is the volume of the box in cubic centimetres?

Introduction / Context:
This is a classic cuboid problem where areas of three mutually perpendicular faces are given, and the task is to determine the volume. Such questions appear often in aptitude exams and test understanding of three dimensional geometry. Instead of giving the edge lengths directly, the problem provides the products of pairs of edges, which requires some algebraic manipulation to extract the volume.

Given Data / Assumptions:
• Let the sides of the cuboid be a, b, and c (in centimetres).• Area of face with sides a and b = ab = 120 cm^2.• Area of face with sides b and c = bc = 72 cm^2.• Area of face with sides c and a = ca = 60 cm^2.• We need the volume V = a * b * c.

Concept / Approach:
The key idea is to multiply the three face areas ab, bc, and ca. This product equals (ab) * (bc) * (ca) = a^2 * b^2 * c^2 = (abc)^2. Taking the square root of this product gives abc, which is the volume. Thus, instead of solving separately for a, b, and c, we can directly calculate the volume from the given face areas. This method is efficient and widely used in similar cuboid problems.

Step-by-Step Solution:
Step 1: Write the face areas in terms of a, b, and c.ab = 120, bc = 72, ca = 60.Step 2: Multiply the three equations.(ab) * (bc) * (ca) = 120 * 72 * 60.Step 3: Recognise that the left side is (abc)^2.(abc)^2 = 120 * 72 * 60.Step 4: Compute the numerical product.120 * 72 * 60 = 518400.Step 5: Take the square root to obtain the volume.abc = sqrt(518400) = 720.Step 6: Therefore, the volume V = 720 cubic centimetres.

Verification / Alternative check:
We can try to factor 518400 to ensure that the square root is correct. Notice that 518400 = 5184 * 100 and 5184 is 72^2. So sqrt(518400) = sqrt(5184) * sqrt(100) = 72 * 10 = 720. This confirms that there was no arithmetic error and validates the computed volume of the cuboid.

Why Other Options Are Wrong:
Option A: 640 cub. cm does not equal the square root of 518400 and would not satisfy the given face areas.Option C: 860 cub. cm is larger than the correct volume and cannot be obtained from the current set of face areas.Option D: 945 cub. cm is unrelated to the calculated value and reflects incorrect handling of multiplication or square rooting.

Common Pitfalls:
Learners sometimes attempt to solve for a, b, and c individually, which is more time consuming and error prone. Another mistake is to add the face areas instead of multiplying them, or to forget that taking the square root is necessary after obtaining (abc)^2. Careful algebraic reasoning and checking the square of the final answer against the intermediate product help avoid these issues.

Final Answer:
The volume of the cuboidal box is 720 cub. cm.