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Q16. A solid cube is cut into eight equal smaller cubes. What is the ratio of the total surface area of these eight smaller cubes and the surface area of the original cube?

एक ठोस घन को आठ बराबर-बराबर छोटे घनों में काटा जाता है। इस आठ छोटे घनों के कुल पृष्ठीय क्षेत्रफल और वास्तविक घन के पृष्ठीय क्षेत्रफल का अनुपात क्या है?


A

8:1

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B

4:1

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C

2:1

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D

1:1

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Solution

The correct option is C

2:1


Method I:

Let one side of the original cube = x

∴ The surface area of the original cube = 6x2

The cube is cut into eight similar cubes as shown below:

As shown in the figure, one side of the smaller cube = x/2

∴ Surface area of one smaller cube = 6(x/2)2 = 3x2/2

∴ Total surface area of 8 similar cubes = 8 × 3x2/2 = 12x2

Hence, required ratio = 12x2/6x2 = 12/6 = 2/1 or 2:1

Method II:

The cube in question is a 2×2×2 cube. Hence, there are 8 smaller cubes and all of them are corner cubes.

We know that out of the total of 6 faces every corner cube has 3 faces exposed. Thus, in case of every corner cube only half of the total surface area is exposed while being a part of the original bigger cube

Thus, required ratio = Total Surface area of 8 individual smaller cubes/Surface area of original cube = Total Surface area of one smaller cube/Exposed Surface area of one smaller cube while being a part of the original bigger cube = 2/1 or 2:1

माना कि वास्तविक घन की एक भुजा = x

वास्तविक घन का पृष्ठीय क्षेत्रफल = 6x2

आठ समान घनों में कटे वास्तविक घन को नीचे दर्शाया गया हैः

आकृति में दिखाए अनुसार, छोटे घन की एक भुजा = x/2

एक छोटे घन की पृष्ठीय क्षेत्रफल = 6(x/2)2 = 3x2/2

8 समान घनों का कुल पृष्ठीय क्षेत्रफल = 8 × 3x2/2 = 12x2

इस प्रकार, आवश्यक अनुपात = 12x2/6x2 = 12/6 = 2/1 या 2:1


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