The areas of three adjacent faces of a solid cuboid are 50 cm², 150 cm² and 75 cm². What is the volume (in cm³) of the cuboid?

Appeared in: DSSSB 13 August 2024

Explanation

The volume of a cuboid is the square root of the product of the areas of its three adjacent faces.
Let the dimensions be l, b, and h. The face areas are A₁=lb, A₂=bh, A₃=hl. The product of these areas is A₁A₂A₃ = (lb)(bh)(hl) = l²b²h² = (lbh)² = V².
Therefore, the volume V = √(A₁ × A₂ × A₃).
Given the areas 50, 150, and 75, their product is 50 × 150 × 75 = 562,500.
The volume is the square root of this product: √562,500 = 750 cm³.

Option B: This value is an arbitrary number and does not result from the correct mathematical formula for the volume.
Option C: This is an incorrect calculation and does not represent the square root of the product of the given areas.
Option D: This value is mathematically incorrect and does not follow the geometric relationship between face areas and volume.

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 concepts in mensuration, which is a key part of the quantitative aptitude section in many competitive exams.
Understanding the relationship between surface area and volume is crucial for solving geometry problems efficiently.
What if the question gave the volume and two face areas? You could find the third face area by rearranging the formula: Area 3 = V² / (Area 1 × Area 2). This demonstrates the versatility of the core formula.

How to Approach the Question

First, identify the given information, which is the areas of three adjacent faces of a cuboid: 50 cm², 150 cm², and 75 cm².
Recall the formula that connects the volume of a cuboid (V) with the areas of its three adjacent faces (A₁, A₂, A₃): V² = A₁ × A₂ × A₃.
Calculate the product of the three given areas: 50 × 150 × 75.
The product is 562,500.
To find the volume, calculate the square root of this product: V = √562,500.
The square root of 562,500 is 750. Therefore, the volume is 750 cm³.

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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