Question

The dimensions of a cuboid are in the ratio 3:2:1. its volume is 1296m³. Find its dimensions​

Answer 1

Given : The dimensions [ Length, Breadth and Height] of a cuboid are in the ratio 3:2:1 and its volume is 1296m³.

To Find : Dimensions of Cuboid [ Length, Breadth and Height]

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$\sf{\bf{ Solution \:of\:Question \::}}$ Let's consider the dimensions [ Length, Breadth and Height] of Cuboid be 3x , 2x & 1x .

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⠀⠀⠀⠀⠀We know that if we are given with the a ratio of Dimensions of Cuboid As, we have already assumed the dimensions according to the given Ratios and we have to find the exact measure of Dimensions of Cuboid with the help of Formula for Volume of Cuboid that is ,

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⠀⠀⠀⠀⠀$\implies {\mathrm { \underline {\bf{ Volume _{( Cuboid)} = l \times b \times h \:or\:lbh\:cu.units\:}}}}$

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⠀⠀⠀⠀⠀Here l is the length of Cuboid, b is the Breadth of Cuboid & h is the Height of Cuboid. And we have given with the value of Volume, and Volume of Cuboid = 1296 m³ , Length (l) = 3x , Breadth (b) = 2x & Height(h) = 1x or x . So by using the Required formula we can find the Value of 'x' :

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⠀⠀⠀⠀⠀⠀⠀⠀⠀$\underline {\sf{\bf{ Now \: By \: Substituting \:the \; Given \: Values \: : }}}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ 1296m^{3} \: = 3x \times 2x \: \times x }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ 1296m^{3} \: = 6x^{2} \times x \: }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ 1296m^{3} \: = 6x^{3} \: }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ \dfrac {1296}{6} \: = x^{3} \: }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ \dfrac {\cancel {1296}^{216}}{\cancel {6}} \: = x^{3} \: }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ x^{3}= 216 \: }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ x= \sqrt [3]{216} \: }}$

⠀⠀⠀⠀⠀$\underline {\boxed{\pink{\sf{ x = 6\:m \: }}}}$

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Therefore,

• Length of Cuboid = 3 x = 3 × 6 = 18 m .
• Breadth of Cuboid = 2x = 2 × 6 = 12 m .
• Height of Cuboid = x = 6 m .

Hence ,

⠀⠀⠀⠀⠀$\underline {\pink{\sf{\therefore \:Hence,\:The\:Dimensions \:of\:Cuboid \:are\:18m\:,12m\:\& \:6m\:.\: }}}$

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Verification:

As , We already know that ,

⠀⠀⠀⠀⠀$\longmapsto {\sf{ L.H.S\:= 1296m^{3} \: }}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\underline {\sf{\bf{ Now \: By \: Finding\:R.H.S \:using\:the \; Finded\: Values \: : }}}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ R.H.S \: = 18 \times 12 \: \times 6 }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ R.H.S \: = 18 \times 12 \: \times 6 }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ R.H.S \: = 216 \times 6 }}$

⠀⠀⠀⠀⠀$\longmapsto {\sf{ R.H.S \: = 1296 m^{3} }}$

Therefore,

⠀⠀⠀⠀⠀$\implies {\sf{ L.H.S\:= R.H.S \: }}$

⠀⠀⠀⠀⠀$\bf\:\:{\dag {\sf{ Hence\:,\:Verified \: }}}$

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