Question

The length, breadth and height of a cuboid are in the ratio 4:3:2 and its total surface area is 468 sq. cm, its actual length, breadth and height will be

Answer 1

${\underline{\underline{\rm{Assumption:}}}}$

• Let ratio be in x from as
• 4x, 3x, 2x.

${\underline{\underline{\rm{Given:}}}}$

• Ratio of Length, Breadth and Height = 4:3:2.
• Total Surface Area = T.S.A = 468 cm².
• If ratio are in x from then
• Length = 4x
• Breadth = 3x
• Height = 2x

${\underline{\underline{\rm{To \: find :}}}}$

• Their Actual Length, Breadth and Height

${\underline{\underline{\rm{Formula \: Required:}}}}$

• T.S.A = 2(lb + bh + lh)
• Here, l implies to Length
• b implies to Breadth
• h implies to Height

${\underline{\underline{\rm{Solution :}}}}$

Here, According to the question we have to make the equation to find the value of x. After that value of will be put into unknown Lenght, Breadth, Height to get their actual values.

● Value of x is

${\bf{\ratio{\longmapsto{T.S.A = 2(lb + bh + lh) }}}} \\ \\ {\sf{\ratio{\longmapsto{468= 2(4x \times 3x + 3x \times 2x + 4x \times 2x) }}}} \\ \\ {\sf{\ratio{\longmapsto{468 = 2 ( 12 {x}^{2} + 6 {x}^{2} + 8 {x}^{2} ) }}}} \\ \\ {\sf{\ratio{\longmapsto{468 = 2(26 {x}^{2} ) }}}} \\ \\ {\sf{\ratio{\longmapsto{ \cfrac{468}{2} = 26 {x}^{2} }}}} \\ \\ {\sf{\ratio{\longmapsto{234 = 26 {x}^{2} }}}} \\ \\ {\sf{\ratio{\longmapsto{ {x}^{2} = \cfrac{234}{26} }}}} \\ \\ {\sf{\ratio{\longmapsto{ {x}^{2} = 9 }}}} \\ \\ {\sf{\ratio{\longmapsto{x = \sqrt{9} }}}} \\ \\ { \underline{ \boxed{ \green{\bf{\ratio{\longmapsto{x = 3 }}}}}}}$

● Now, Length, Breadth and Height are :-

$\large{\sf{\ratio{\longmapsto{Length = 4x }}}}\\\large{\sf{\ratio{\longmapsto{Length = 4(3) }}}}\\\large{\underline{\boxed{\red{\sf{\ratio{\longmapsto{Length = 12 \: cm}}}}}}}$

____________________

$\large{\sf{\ratio{\longmapsto{Breadth = 3x }}}}\\\large{\sf{\ratio{\longmapsto{Breadth = 3(3) }}}}\\\large{\underline{\boxed{\red{\sf{\ratio{\longmapsto{Breadth = 9 \: cm}}}}}}}$

____________________

$\large{\sf{\ratio{\longmapsto{Height = 2x }}}}\\\large{\sf{\ratio{\longmapsto{Height = 2(3) }}}}\\\large{\underline{\boxed{\red{\sf{\ratio{\longmapsto{Height = 6 \: cm}}}}}}}$

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● Diagram

$\setlength{\unitlength}{0.74 cm}\begin{picture}\thicklines\put(5.6,5.4){\bf }\put(11.1,5.4){\bf }\put(11.2,9){\bf }\put(5.3,8.6){\bf }\put(3.3,10.2){\bf }\put(3.3,7){\bf }\put(9.25,10.35){\bf }\put(9.35,7.35){\bf }\put(3.5,6.1){\sf 9\:cm}\put(7.7,6.3){\sf 12\:cm}\put(11.3,7.45){\sf 6\:cm}\put(6,6){\line(1,0){5}}\put(6,9){\line(1,0){5}}\put(11,9){\line(0,-1){3}}\put(6,6){\line(0,1){3}}\put(4,7.3){\line(1,0){5}}\put(4,10.3){\line(1,0){5}}\put(9,10.3){\line(0,-1){3}}\put(4,7.3){\line(0,1){3}}\put(6,6){\line(-3,2){2}}\put(6,9){\line(-3,2){2}}\put(11,9){\line(-3,2){2}}\put(11,6){\line(-3,2){2}}\end{picture}$

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● VERIFICATION :-

》T.S.A = 2(lb + bh + lh)

》468 = 2(12×9 + 9×6 + 12×6)

》468 = 2(108 + 54 + 72)

》468 = 2(234)

》468 = 468

Here, LHS = RHS, So, our answers are correct.

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● More Formulas related to Cuboid :-

• Lateral surface area of a cuboid = 2(l + b) × h
• Volume of a'cuboid = l × b × h
• Diagonal of the cuboid =√( l²+ b² +h²)
• Perimeter of cuboid = 4 (length + breadth + height

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