Math, asked by MяMαgıcıαη, 1 month ago

The dimensions of a cuboid are in the ratio of 2:3:4 and its total surface area is 280m^2. Find the dimensions.
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Answers

Answered by SuitableBoy

151

$\large{\bf{\underbrace{\underline{Required~Answer:-}}}}$

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$\frak{Given}\begin{cases}\sf{Dimensions\:of\:a\:Cuboid=\bf{2:3:4}}\\\sf{Total\:Surface\:Area=\bf{280\:m^2}}\end{cases}$

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$\underline{\bf{\green\dag\:To~Find:}}$

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The dimensions of the cuboid.

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$\underline{\bf{\green\dag\:Solution:}}$

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» In this question, we are given with the ratio of the dimensions of a cuboid, we would first make the ratios in numbers, by multiplying by a constant (Let's say x).

» Then, Using the dimensions and the formula for finding the Total Surface Area, we would find the constant 'x' .

» After finding the constant, we would multiply it with the given ratio so as to get the dimensions of the cuboid.

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◑ Let the ratios be 2x , 3x & 4x .

It means :

Length = 4x
Breadth = 3x
Height = 2x

We know,

$\odot\;\boxed{\sf T. S. A_{cuboid}=2(lb+bh+lh) }$

In this formula,

l = length of the cuboid.
b = breadth of the cuboid.
h = height of the cuboid.

Now, put the value of TSA , l , b & h in the formula.

$\colon \rarr \sf \: \cancel{280} \: {m}^{2} = \cancel2 \{(4x \times 3x) + (3x \times 2x) + (2x \times 4x) \} \\ \\ \colon \rarr \sf \: 140 \: {m}^{2} = 12x {}^{2} + 6x {}^{2} + 8x {}^{2} \\ \\ \colon \rarr \sf \: \cancel{140} \: {m}^{2} = \cancel{26} \: {x}^{2} \\ \\ \displaystyle \colon \rarr \sf \: {x}^{2} = \dfrac{70}{13} \: {m}^{2} \\ \\ \colon \rarr \sf \: {x}^{2} = 5.38 \: {m}^{2} \\ \\ \colon \dashrightarrow \boxed{ \frak{ \green{x = 2.32 \: m}}}$

Now,

$\colon \leadsto \sf \: length = 4x = 4 \times 2.32 \\ \\ \colon \leadsto \underline{ \boxed{ \frak{ \purple{length = \sf{9.28 }\: \frak m}}}}$

$\colon \leadsto \sf \: breadth = 3x = 3 \times 2.32 \\ \\ \colon \leadsto \underline{ \boxed{ \frak{ \pink{breadth = \sf 6.96 \: \frak m}}}}$

$\colon \leadsto \sf \: height = 2x = 2 \times 2.32 \\ \\ \colon \leadsto \underline{ \boxed{ \frak{ \red{height = \sf4.62 \: \frak m}}}}$

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$\therefore\;\underline{\sf Dimensions\:(Cuboid) =\bf{9.28m\times 6.96m\times 4.62m.}}\\$

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◆ Note :– The values here are aprox, u may keep them is the form of square root also.

Answered by Anonymous

Given:

The dimensions of a cuboid are in the ratio of 2 :3:4

and its total surface area is 280m²

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To Find:

The dimensions of the cubiod.

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Solution:

❍ Now, as said that the dimensions are in the ratio 2 : 3: 4

let's consider the dimensions of the cubiod as 2x,3x,4x

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◐ here the value of the total surface area of the cubiod is given, so with the help of it and taking the dimensions 2x,3x and 4x let's frame an equation and find out the measurements of the dimensions

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$\pink{\underline{\mathfrak{As\:we\:know\:that;}}}$

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${\longrightarrow }\:\blue{ \underline{ \boxed{ \pink{ \mathfrak{ the \: t.s.a \: of \: a \: cuboid = 2(lb + bh + hl}}}} \bigstar}$

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❍ let's substitute the values now,

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${ : \implies} \sf \: t.s.a = 2(lb + bh + hl) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf280 {m}^{2} = 2(2x \times 3x + 3x \times 4x + 4x \times 2x) \\ \\ \\ \\ { : \implies} \sf \: 280 {m}^{2} = 2(6 {x}^{2} + 12 {x}^{2} + 8 {x}^{2} )\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \cancel\frac{280}{2} = 26 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: 140 = 26 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf {x}^{2} = 5.38\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies} \sf \: x = \sqrt{5.38} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \\ { : \implies}\: \large{ \underline{\boxed{ \orange{\tt {x = 2.31}}}}} \orange\bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$