Question

How many wooden cubical blocks of side 15 cm can be cut from a log of wood of size 6 m by 45 cm by 30 cm ?​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the Concept of Volume of Cube and Cuboid has been used. We see here firstly we will find the volume of each cube. Then we will find the volume of the Cuboidal Log of Wood. We know that Volume never changed since its the space occupied by matter. So sum of volume of number of cubes formed will be equal to volume of the wood.

Let's do it !!

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★ Formula Used :-

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

$\\\;\boxed{\sf{\purple{Volume\;of\;Cuboid\;=\;\bf{Length\;\times\;Breadth\;\times\;Height}}}}$

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★ Solution :-

Given,

» Side of each cuboidal block = 15 cm

» Dimensions of wooden block = 6 m × 45 cm × 30 cm = 600 cm × 45 cm × 30 cm

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~ For the Volume of Each Cuboidal Block ::

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

$\\\;\:\sf{:\rightarrow\;\;Volume\;of\;Cube\;=\;\bf{(15)^{3}}}$

$\\\;\:\underline{\bf{:\rightarrow\;\;Volume\;of\;Cubical\;Piece\;=\;\bf{\red{3375\;\;cm^{3}}}}}$

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~ For the Volume of the Wooden Log ::

$\\\;\;\sf{:\Longrightarrow\;\;Volume\;of\;Cuboid\;=\;\bf{Length\;\times\;Breadth\;\times\;Height}}$

$\\\;\;\sf{:\Longrightarrow\;\;Volume\;of\;Cuboid\;=\;\bf{600\;\times\;45\;\times\;30}}$

$\\\;\:\underline{\bf{:\rightarrow\;\;Volume\;of\;Wooden\;Log\;=\;\bf{\red{810000\;\;cm^{3}}}}}$

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~ For the Number of Cuboidal Blocks ::

This is given as,

$\\\;\;\sf{:\mapsto\;\;Number\;of\;Cubical\;Blocks\;=\;\bf{\dfrac{Volume\;of\;Wooden\;Log}{Volume\;of\;cubical\;piece}}}$

$\\\;\;\sf{:\mapsto\;\;Number\;of\;Cubical\;Blocks\;=\;\bf{\dfrac{810000}{3375}}}$

$\\\;\;\sf{:\mapsto\;\;Number\;of\;Cubical\;Blocks\;=\;\bf{\blue{240\;\;pieces}}}$

$\\\;\underline{\boxed{\tt{Number\;\;of\;\;Cubical\;\;Blocks\;=\;\bf{\green{240\;\;pieces}}}}}$

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★ More to know :-

$\\\;\sf{\pink{\leadsto\;\;Volume\;of\;Sphere\;=\;\dfrac{4}{3}\;\pi r^{3}}}$

$\\\;\sf{\pink{\leadsto\;\;Volume\;of\;Hemisphere\;=\;\dfrac{2}{3}\;\pi r^{3}}}$

$\\\;\sf{\pink{\leadsto\;\;Volume\;of\;Cylinder\;=\;\pi r^{2}h}}$

$\\\;\sf{\pink{\leadsto\;\;Volume\;of\;Cone\;=\;\dfrac{1}{3}\;\pi r^{2}h}}$

$\\\;\sf{\pink{\leadsto\;\;Volume\;of\;Hollow\;Cylinder\;=\;\pi (R^{2}\;-\;r^{2})h}}$

Answer 2

${ \bold { \underline{\large\purple{ Answer : - }}}} \:$

Wooden cubical block :-

$\setlength{\unitlength}{4mm}\begin{picture}(10,6)\thicklines\put(0,1){\line(0,1){10}}\put(0,1){\line(1,0){10}}\put(10,1){\line(0,1){10}}\put(0,11){\line(1,0){10}}\put(0,11){\line(1,1){5}}\put(10,11){\line(1,1){5}}\put(10,1){\line(1,1){5}}\put(0,1){\line(1,1){5}}\put(5,6){\line(1,0){10}}\put(5,6){\line(0,1){10}}\put(5,16){\line(1,0){10}}\put(15,6){\line(0,1){10}}\put(4.6,-0.5){\bf\large 15 cm}\put(13.5,3){\bf\large 15 cm}\put(-4,5.8){\bf\large 15 cm}\end{picture}$

→ Given :-

• Side of cubical block → 15 cm.

Log of wood :-

$\setlength{\unitlength}{0.74 cm}\begin{picture}\thicklines\put(5.6,5.4){\bf A}\put(11.1,5.4){\bf B}\put(11.2,9){\bf C}\put(5.3,8.6){\bf D}\put(3.3,10.2){\bf E}\put(3.3,7){\bf F}\put(9.25,10.35){\bf H}\put(9.35,7.35){\bf G}\put(3.5,6.1){\sf 45\:cm}\put(7.7,6.3){\sf 6\:m}\put(11.3,7.45){\sf 30\:cm}\put(6,6){\line(1,0){5}}\put(6,9){\line(1,0){5}}\put(11,9){\line(0,-1){3}}\put(6,6){\line(0,1){3}}\put(4,7.3){\line(1,0){5}}\put(4,10.3){\line(1,0){5}}\put(9,10.3){\line(0,-1){3}}\put(4,7.3){\line(0,1){3}}\put(6,6){\line(-3,2){2}}\put(6,9){\line(-3,2){2}}\put(11,9){\line(-3,2){2}}\put(11,6){\line(-3,2){2}}\end{picture}$

Also Given :-

• Log of wood of size 6 m by 45 cm by 30 cm.

${ \bold { \underline{\large\pink{Solution : - }}}} \:$

Lets we find the volume of cubical Block :-

Here ,we Have

So, we are going to use the formula of volume of cube.

Edge of each cube = 15 cm

• Volume of Cube = (Edge)³

⇒Volume of Cube = (15)³

⇒ V = 3375cm²

Now,

Volume of the Wooden Log which is cuboidal in shape.

• Volume of Cuboid = Length × Breadth × Height

Therefore,

⇒ Length given as " 6m "

⇒ Breadth given as " 45 cm "

⇒ Height given as " 30 cm "

Note :- First check the unit of all the values otherwise this silly mistake will make your whole answer wrong.

Changing the given length into cm = 600 cm

Now, putting up the value.

Volume = 600 × 45 × 30

Volume of Log = 810000cm²

Let " n " be the number of cubes that can be cut off from the given cuboid.

Then ,

Volume of the Cuboid = Number of cubes × Volume of each cube.

⇒ V = n × v

Where,

" V " = Volume of Cuboid

" v " = Volume of each cube

" n " = Number Of cubes.

→ n = V/v

→ n = 810000/3375

→ n = 240 .

Hence, the required number of cubes that can be cut from the given cuboid is 240.

⠀⠀⠀⠀★ Working Rule

Conversion of Solids from One shape to another.

~ We often need to convert a solid from one shape to another. For example , a metallic sphere is melted and recast into a wire of a cylindrical shape. Now the key concept that we have to remember during such conversions of solid is that the volume of the new solid of a new shape will be the same as the volume of the original solid of different shape.

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