The sum of the length, breadth, and height of a cuboid is 20 cm, and the length of its diagonal is 12 cm. Find the surface area of the cuboid.

Appeared in: NHM UP Staff Nurse-2023 shift 1st

Explanation

The solution relies on the algebraic identity (l + b + h)² = l² + b² + h² + 2(lb + bh + hl).
The term (l + b + h) is the given sum of dimensions (20 cm).
The term l² + b² + h² is the square of the given diagonal length (12² = 144 cm²).
The term 2(lb + bh + hl) represents the total surface area of the cuboid.
Substituting the given values: (20)² = 144 + Surface Area, which simplifies to 400 = 144 + Surface Area.
Solving for the surface area gives 400 - 144 = 256 cm².

Option A: This value is incorrect and does not result from the correct application of the formula.
Option B: This value is incorrect. It may arise from a calculation error when subtracting 144 from 400.
Option D: This value is incorrect. It may arise from a simple arithmetic mistake during the final subtraction step.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of algebraic identities to solve mensuration problems involving cuboids as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical reasoning and the ability to apply abstract formulas to solve a concrete problem.
Understanding how different geometric properties are related through algebraic identities is a key problem-solving skill applicable in various scientific and technical fields.
What if the diagonal was 10 cm? Then l² + b² + h² = 100. The surface area would be 400 - 100 = 300 cm².

How to Approach the Question

First, identify the geometric shape (cuboid) and the given parameters: the sum of its dimensions (l + b + h = 20) and the length of its diagonal (d = 12).
Next, identify the quantity you need to find: the total surface area (TSA).
Recall the standard formulas for a cuboid: Diagonal d = √(l² + b² + h²) and TSA = 2(lb + bh + hl).
Look for a relationship that connects the given information to the required value. Recognize that the algebraic identity (l + b + h)² = l² + b² + h² + 2(lb + bh + hl) links all three components.
Substitute the given values into this master equation: (20)² = (12)² + TSA.
Solve the equation: 400 = 144 + TSA, which gives TSA = 256 cm². Select the corresponding option.

Concept Tested & Keywords

Concept Tested: Application of algebraic identities to solve mensuration problems involving cuboids.
Stem keywords: cuboid, sum of length, breadth, and height, diagonal, surface area
Lead-in keywords: Find
Negative lead-in flag: false

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Practise the full NHM UP Staff Nurse-2023 shift 1st

Attempt every question from this paper in a timed mock, then review the full solution for each one.