Question

A box of a beauty soap is 6cm by 3cm by 3cm . If such soap are to be packed inside a bigger box 30cm by 25cm by 25cm in dimensions, what maximum number of soap can the bigger box hold?

Answer 1

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

Here the concept of Volume of Cuboid have been used. We see that we are given two cuboidal solids that is beauty soap and the box in which they are kept. So firstly we can find the volume of the box and then volume of each soap. The volume occupied by number of soaps, will be equal to the volume of box. Thus we can find out our answer.

Let's do it !!

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★ Formula Used :-

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

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★ Solution :-

Given,

» Dimensions of Beauty Soap = 6 cm × 3 cm × 3 cm

» Dimensions of box of soap = 30 cm × 25 cm × 25 cm

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~ For the Volume of each Soap ::

We know that,

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

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Volume\;of\;Cuboid_{\green{(Soap)}}\;=\;\bf{6\:\times\:3\:\times\:3}}$

$\\\;\bf{:\Longrightarrow\;\;Volume\;of\;Cuboid_{\green{(Soap)}}\;=\;\bf{\orange{54\;\;cm^{3}}}}$

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~ For the Volume of Soap Box ::

We know that,

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

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Volume\;of\;Cuboid_{\blue{(Box)}}\;=\;\bf{30\:\times\:25\:\times\:25}}$

$\\\;\bf{:\Longrightarrow\;\;Volume\;of\;Cuboid_{\blue{(Box)}}\;=\;\bf{\orange{18750\;\;cm^{3}}}}$

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~ For the number of soaps that bigger box can hold ::

This is given as,

$\\\;\sf{\mapsto\;\;\gray{No.\;of\;Soaps\;=\;\bf{\dfrac{Volume\;of\;Box}{Volume\;of\;Soap}}}}$

By applying values, we get

$\\\;\sf{\mapsto\;\;No.\;of\;Soaps\;=\;\bf{\dfrac{18750}{54}}}$

$\\\;\sf{\mapsto\;\;No.\;of\;Soaps\;=\;\bf{\red{347.22\;\approx\;347}}}$

Here we took approximation since number of pieces can't be in decimal.

Hence, maximum number of soaps that box can hold = 347 soaps.

$\\\;\underline{\boxed{\tt{Maximum\;number\;of\;soaps\;=\;\bf{\purple{347\;soaps}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;CSA\;of\;Cuboid\;=\;2(Length\:+\:Breadth)\:\times\:Height}$

$\\\;\sf{\leadsto\;\;TSA\;of\;Cuboid\;=\;2(LB\;+\;BH\;+\;LH)}$

$\\\;\sf{\leadsto\;\;Diagon\;of\;Cuboid\;=\;\sqrt{(L^{2}\;+\;B^{2}\;+\;H^{2})}}$

• Cuboid has 6 faces.

• Cuboid has 12 edges.

Answer 2

${\large{\bold{\rm{\underline{Let's \; understand \; the \; question \; 1^{st}}}}}}$

✦ This question says that there is a box of beauty soaps of dimensions ( length , breadth and height ) ; 6 cm , 3 cm and 3 cm respectively. And there soaps are packed inside a bigger box of dimensions ( length , breadth and height ) ; 30 cm , 25 cm and 25 cm respectively. (Using concept - Volumes) And at last we have to find the number of soap can be packed in the bigger box. Let's solve it.

${\large{\bold{\rm{\underline{Given \; that}}}}}$

✠ Length of soap = 6 cm

✠ Breadth of soap = 3 cm

✠ Height of soap = 3 cm

✠ Length of box = 30 cm

✠ Breadth of box = 25 cm

✠ Height of box = 25 cm

${\large{\bold{\rm{\underline{To \; find}}}}}$

✠ The number of soap can be packed in the bigger box.

${\large{\bold{\rm{\underline{Solution}}}}}$

✠ The number of soap can be packed in the bigger box = 347

${\large{\bold{\rm{\underline{Using \; concepts}}}}}$

✠ Formula to find volume.

(According to the question,) Formula to find number of soaps can be packed in the bigger box.

${\large{\bold{\rm{\underline{Using \; formulas}}}}}$

✠ Volume = L × B × H

$\: \: \: \: \: \: \: \: \: \: \: \: \:{\sf{Where,}}$

✦ L denotes length

✦ B denotes breadth

✦ H denotes height

✠ Number of soaps = Volume of box / Volume of soap

${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

~ Using above data firstly let us find the volume of the beauty soaps..!

➨ Volume of soap = L × B × H

➨ Volume of soap = 6 × 3 × 3

➨ Volume of soap = 6 × 9

➨ Volume of soap = 54 cm³

${\pink{\frak{Henceforth, \: volume \: of \: soap \: is \: 54 \: cm^{3}}}}$

~ Now using above data again, let's find the volume of box..!

➨ Volume of box = L × B × H

➨ Volume of box = 30 × 25 × 25

➨ Volume of box = 750 × 25

➨ Volume of box = 18750 cm³

${\pink{\frak{Henceforth, \: volume \: of \: box \: is \: 18750 \: cm^{3}}}}$

~ Now let's find the number of soaps can the bigger box hold.

➨ Number = Volume of box / Volume of soap

➨ Number = 18750 cm³ / 54 cm³

➨ Number = 347.22

~ But we haven't take the number of soaps in approximately so the number of soaps are 347.

${\pink{\frak{Henceforth, \: number \: of \: soap \: is\: 347}}}$

${\large{\bold{\rm{\underline{Additional \: knowledge}}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto TSA \: of \: cuboid \: = \: 2(l \times b + b \times h + l \times h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto LSA \: of \: cuboid \: = \: 2h(l+b)}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cuboid \: = \: L \times B \times H}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Diagonal \: of \: cuboid \: = \: \sqrt 3l}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Perimeter \: of \: cuboid \: = \: 12 \times Sides}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cylinder \: = \: \pi r^{2}h}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Surface \: area \: of \: cylinder \: = \: 2 \pi rh + 2 \pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Lateral \: area \: of \: cylinder \: = \: 2 \pi rh}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Height \: of \: cylinder \: = \: \dfrac{v}{\pi r^{2}}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Radius \: of \: cylinder \: = \: \dfrac{v}{\pi h}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto LSA \: of \: cube \:= \: 4(side)^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Volume \: of \: cube \: = \: (side)^{3}}}}$

$\; \; \; \; \; \; \;{\sf{\bold{\leadsto Perimeter \: of \: cube \: = \: 4(l+b+h)}}}$