Question

The dimensions of a cuboid are in the ratio 3 : 2 : 1 and the total surface area is 198 sq. cm. Find its volume.

5 points

Your answer


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Answer 1

$\frak{Given}\begin{cases}\sf{\: Ratio \ of \ dimensions \ of \ cuboid \ is \: \bf{3:2:1}}\\\sf{Total \ surface \ are \ is \ \bf{198 \ cm^2}}\end{cases}$

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We've to find out the volume of cuboid.

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☯ Let's consider that the Length, Breadth and Height of the cuboid be 3x, 2x & 1x.

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$\dag\;{\underline{\frak{As \ we \ know \ that,}}}$

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$\star\;\boxed{\sf{\pink{Total \ surface \ Area_{\:(cuboid)} = 2(lb + bh + hl)}}}$

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• Total surface area is 198 cm².

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$\dag\;{\underline{\frak{Substituting \ values \ :}}}$

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$:\implies\sf 198 = 2(lb + bh + hl) \\\\\\:\implies\sf 198 = 2(3x \times 2x + 2x \times x + x \times 3x) \\\\\\:\implies\sf 198 = 2(6x^2 + 2x^2 + 3x^2) \\\\\\:\implies\sf 198 = 2(11 x)^2 \\\\\\:\implies\sf 198 = 22x^2\\\\\\:\implies\sf x^2 = \cancel\dfrac{198}{22}\\\\\\:\implies\sf x^2 = 9\\\\\\:\implies{\underline{\boxed{\frak{\purple{x = 3}}}}}$

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Now,

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• Length of cuboid, 3x = 3(3) = 9

• Breadth of cuboid, 2x = 2(3) = 6

• Height of cuboid, x = 3

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$\star\;\boxed{\sf{\pink{Volume \ of \ cuboid = l \times b \times h }}}$

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$:\implies\sf Volume_{\:(cuboid)} = 9 \times 6 \times 3 \\\\\\:\implies\sf Volume_{\:(cuboid)} = 54 \times 3 \\\\\\:\implies{\underline{\boxed{\frak{\purple{ Volume_{\:(cuboid)} = 162 \ cm^3}}}}}$

$\therefore\:{\underline{\sf{Volume \ of \ cuboid \ is\: \bf{162~cm^3}.}}}$

Answer 2

Given:

• The Dimensions of a cuboid are in the ratio 3 : 2 :1
• Te total surface area is 198 cm²

To Find:

• Volume of cuboid

Let,

⊱ the length of cuboid be 3x

⊱ breadth be 2x

⊱ height be x

We know that:

$\sf\underline\orange{ Total\:surface\:area\:of\:cuboid\:=\:2\:(lb + bh + hl )}$

Put the given values:

$\sf\implies198=2(3x\times2x+2x\times\:x+x\times3x)$

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$\sf\implies2(6x^2+2x^2+3x^2)=198$

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$\sf\implies(6x^2+2x^2+3x^2)=\dfrac{198}{2}$

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$\sf\implies(6x^2+2x^2+3x^2)=99$

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$\sf\implies(11x^2)=99$

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$\sf\implies\:x^2=9$

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$\sf\implies\:x=\sqrt{9}$

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$\sf\implies\:x=3$

thus,

Length = 3x = 9 cm

Breadth = 2x = 6cm

Height= x = 3 cm

Now,

$\sf\underline\red{volume\:of\: cuboid\:=length\:\times\:Height\:\times\:Breadth}$

$\sf=9\times6\times3$ $\sf=162cm^3$

Hence,

Volume of the cuboid = 162²