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MathematicsMensuration
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A solid cube of side 6 cm is melted and recast into small cubes of side 2 cm. How many such small cubes can be formed?

A

8

B

16

C

24

D

27

Correct Answer

ЁЯУР MA тАв Math Concept & PrincipleMathematicsMensuration
Option D

27

Quick Summary:

When a solid object is melted and recast into smaller objects of the same material, the total volume remains conserved. By dividing the volume of the large cube by the volume of a single small cube, we find that exactly 27 small cubes can be formed.

ЁЯФмSCScience SolutionConcept & Principle
ЁЯТб Explanation

When a solid object is melted and recast into smaller objects of the same material, the total volume remains conserved. By dividing the volume of the large cube by the volume of a single small cube, we find that exactly 27 small cubes can be formed.

ЁЯТб Everyday Analogy (Real-World Intuition)

Imagine melting a large block of chocolate to pour into smaller cubic molds; the total volume of chocolate stays constant, and the number of pieces depends on how many times smaller the new volume is.

ЁЯФв Key Formulas

V=s3V = s^3V=s3 тАФ Volume of a cube where sss is the side length

N=(slargessmall)3N = \left(\frac{s_{\text{large}}}{s_{\text{small}}}\right)^3N=(ssmallтАЛslargeтАЛтАЛ)3 тАФ Number of recast similar solid objects

тЪЩя╕П Working Principle

The principle of conservation of volume dictates that the total volume of matter remains constant during a geometric transformation or phase change, assuming no material is lost. The volume of any cube is given by the formula V=s3V = s^3V=s3, where sss is the side length. Therefore, the number of small cubes NNN is determined by the ratio of their volumes: N=slarge3ssmall3=(62)3=33=27N = \frac{s_{\text{large}}^3}{s_{\text{small}}^3} = \left(\frac{6}{2}\right)^3 = 3^3 = 27N=ssmall3тАЛslarge3тАЛтАЛ=(26тАЛ)3=33=27.

ЁЯУМ Key Points
  • тЦ╕

    Volume is conserved during melting and recasting processes.

  • тЦ╕

    Scaling factor for volume increases cubically with linear dimensions.

ЁЯЫая╕П Applications / Uses
  • тЦ╕

    Metal casting and foundry operations in metallurgy

  • тЦ╕

    Injection molding in polymer manufacturing

ЁЯУД Additional Information
  • тЦ╕

    Volume is measured in cubic units such as cm3\text{cm}^3cm3.

  • тЦ╕

    Option A (8) incorrectly uses a linear ratio or miscalculates as 232^323, while Option C (24) is a common calculation trap.

ЁЯУК Diagram / Illustration
Cube Melting and Recasting Problem Solution1Given Data: Large and Small CubesLarge cube side (SL) = 6 cm, Small cube side (s) = 2 cm2Formula: Conservation of VolumeVolume of Cube = s┬│, Number of small cubes N = V_large / V_small3Calculation StepN = (6┬│ / 2┬│) = (6 / 2)┬│ = 3┬│ = 274Final Count of Small CubesExact number of small cubes formed = 27Correct Option: D) 27 (Formula: N = (SL / s)┬│)
тЬЕ

D is correct тАФ 27 small cubes are formed because the ratio of the volume of the large cube to the small cube is 216/8=27216 / 8 = 27216/8=27.

Core Concepts Used
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conservation of volume cube volume formula geometric scaling
ЁЯТб EXAM TIP

Always verify whether a mensuration problem requires scaling of length (sss), surface area (s2s^2s2), or volume (s3s^3s3).

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