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.​

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

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