The areas of three adjacent faces of a solid cuboid are 297 cm2, 33cm2 and 196cm2. What is the volume (in cm}3 of the cuboid?

Appeared in: DSSSB 12 August 2024

Explanation

The volume (V) of a cuboid is related to the areas of its three adjacent faces (A₁, A₂, A₃) by the formula V² = A₁ × A₂ × A₃.
Given the areas 297 cm², 33 cm², and 196 cm², their product is 297 × 33 × 196 = 1,920,996.
The volume is the square root of this product: V = √1,920,996 = 1386 cm³.
An easier calculation method involves factorization: V = √( (9×33) × 33 × 14² ) = √(3² × 33² × 14²) = 3 × 33 × 14 = 1386.

Option A: This is an incorrect calculation. The square of 1310 is 1,716,100, which is not equal to the product of the face areas (1,920,996).
Option B: This is a miscalculation. The square of 1307 is 1,708,249, which does not match the required product of 1,920,996.
Option C: This value is incorrect. The square of 1173 is 1,375,929, which is not equal to the product of the face areas (1,920,996).

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Calculating the volume of a cuboid from the areas of its adjacent faces as background academic context rather than a clinical decision trigger.
This question tests fundamental mathematical and spatial reasoning skills, which are part of the general aptitude section in many nursing entrance exams.
While not a direct clinical skill, proficiency in basic mathematics is essential for tasks like medication dosage calculation, fluid balance monitoring (I/O charting), and interpreting data from medical devices.
What if? If one of the areas was different, say 66 cm² instead of 33 cm², the entire calculation would change. The new volume would be √(297 × 66 × 196) = 1960.2, demonstrating how a change in one parameter affects the final result.

How to Approach the Question

First, identify the given information: the areas of three adjacent faces of a cuboid (A₁, A₂, A₃).
Recall the relationship between these areas and the cuboid's volume (V). Let the dimensions be l, b, h. The areas are A₁=lb, A₂=bh, A₃=hl.
Multiply the three areas: A₁ × A₂ × A₃ = (lb) × (bh) × (hl) = l²b²h² = (lbh)².
Recognize that V = lbh, so the product of the areas is equal to the square of the volume (V²).
Calculate V by taking the square root of the product of the given areas: V = √(297 × 33 × 196).
To simplify the calculation, use factorization: √( (9×33) × 33 × 14² ) = √(3² × 33² × 14²) = 3 × 33 × 14 = 1386.

Concept Tested & Keywords

Concept Tested: Calculating the volume of a cuboid from the areas of its adjacent faces.
Stem keywords: cuboid, adjacent faces, area, volume
Lead-in keywords: What is the volume

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