Question

Q5. The dimensions of a metal block are 2.25 m by 1.5 m by 27 m. It is melted ad recast into cubes, each of the side 45 cm. How many cubes are formed?

Answer 1

Answer :-

$\: \: \boxed{\boxed{\rm{\green{\mapsto \: \: \: Before \; solving, \; refer \; here \; please \; !!}}}}$

Here the concept of the equality in Volumes has been used. We know that its given, the cuboidal metal block had been recasted into many cubes and we have to find number of cubes formed. We know that finally number of cubes formed multiplied to volume of each cube will give us the volume oof the initial Cuboidal block. Because volume is the amount of matter which cannot be increased or decreased. Let's do it !!

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★ Formula Used :-

$\: \: \large{\boxed{\boxed{\sf{\purple{Volume \: of \: Cuboid \: = \: \bf{Length(L) \: \times \: Breadth(B) \: \times \: Height(H)}}}}}}$

$\: \large{\boxed{\boxed{\sf{\purple{Volume \: of \: Cube \: = \: \bf{(Side)^{3}}}}}}}$

$\: \large{\boxed{\boxed{\sf{\purple{Number \: of \: Cubical \: pieces \: formed \: = \: \bf{\dfrac{Volume \: of \: Initial \: Cuboid}{Volume \: of \: each \: Cube \: pieces}}}}}}}$

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★ Question :-

The dimensions of a metal block are 2.25 m by 1.5 m by 27 m. It is melted ad recast into cubes, each of the side 45 cm. How many cubes are formed?

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★ Solution :-

Given,

» Dimensions of metallic cuboidal block = 2.25 m × 1.5 m × 27 m

= 225 cm × 150 cm × 2700 cm

» Side of each Cube piece = 45 cm

Now all the units are converted into same.

Then, according to the question :-

~ Volume of Initial Cuboidal Block :-

$\: \: \large{\sf{\Longrightarrow \: \: \: Volume \: of \: Cuboid \: = \: \bf{Length(L) \: \times \: Breadth(B) \: \times \: Height(H)}}}$

➣ Volume of Cuboid = 225 cm × 150 cm × 2700 cm

➣ Volume of Cuboid = 91,125,000 cm³

$\: \\ \: \large{\boxed{\boxed{\tt{Volume \; of \; Initial \; Cuboid \; = \; \bf{\pink{91,125,000 \; cm^{3}}}}}}}$

~ Volume of each Cuboidal Pieces :-

$\: \large{\sf{\Longrightarrow \: \: \: Volume \: of \: Cube \: = \: \bf{(Side)^{3}}}}$

➣ Volume of Cube = (45 cm)³

➣ Volume of Cube = 91,125 cm³

$\: \\ \: \large{\boxed{\boxed{\tt{Volume \; of \; Initial \; Cuboid \; = \; \bf{\pink{91,125 \; cm^{3}}}}}}}$

~ Number of cubical pieces formed :-

$\: \\ \large{\sf{\Longrightarrow \: \: \: Number \: of \: Cubical \: pieces \: formed \: = \: \bf{\dfrac{Volume \: of \: Initial \: Cuboid}{Volume \: of \: each \: Cube \: pieces}}}}$

$\: \qquad \qquad \large{\rm{\longmapsto \: \: Number \; of \; Cubical \; pieces \; formed \; = \; \bf{\dfrac{91125000 \; \not{cm^{3}}}{91125 \: \not{cm^{3}}} \: = \: \underline{\underline{\orange{1000}}}}}}$

➣ Number of cubical pieces = 1000

$\: \: \large{\boxed{\boxed{\tt{Number \; of \; Cubical \; pieces \; formed \; = \; \bf{\pink{\underline{1000}}}}}}}$

$\: \\ \underline{\underline{\sf{\leadsto \: \: \: Thus \: number \: of \: cubical \: pieces \: formed \: out \: of \: recasting \: the \: metallic \: block \: are \: \boxed{\bf{\blue{1000}}}}}}$

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$\: \: \large{\underline{\underline{\orange{\it{\mapsto \: \: \: Confused?, \; \: Don't \; \: worry \: \; let's \: \; verify \: \; it \: :-}}}}}$

For verification we need to simply apply the value we got into the equation we formed.

Then, let n be the number of cubical pieces formed.

=> n × Volume of cubical pieces = Volume of initial cuboid

=> 1000 × 45 × 45 × 45 = 225 × 150 × 2700

=> 91125000 = 91125000

Clearly, LHS = RHS

Here the condition satisfies so our answer is correct.

Hence, Verified.

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$\: \qquad \qquad \large{\underbrace{\sf{\leadsto \: \: \: Aid \; to \; Memory \: :}}}$

• Volume of Cylinder = πr²h

• Volume of Cone = ⅓ × πr²h

• Volume of Sphere = 4/3 × πr³

• Volume of Hemisphere = ⅔ × πr³

• LSA of Cube = 4 × (Side)²

• TSA of Cube = 6 × (Side)²

• LSA of Cuboid = 2 × (Length + Breadth) × Height

• CSA of Cylinder = 2πrh

• CSA of Cone = πrl

• TSA of Cylinder = 2πrh + 2πr²

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Answer 2

Given :

The dimensions of a metal block are 2.25 m by 1.5 m by 27 m. It is melted ad recast into cubes, each of the side 45 cm.

To find :

How many cubes are formed

Solution :

Given , dimensions of block = 2.25 * 1.5 * 27 m

∴ Volume of block = 2.25 * 1.5 * 27

∴ Volume of block = 91.125 m³

Now side of cube = 45 cm = 45/100 = 0.45 m

∴ Volume of cube = Side³

⇒ Volume of cube = (0.45)³

⇒ Volume of cube = 0.091125 cm³

Now let the number of cubes be formed as n

⇒ n = Volume of block/Volume of cube

⇒ n = 91.125/0.091125

⇒ n = 91125000/91125

⇒ n = 1000

Therefore,

Number of cubes are formed = 1000