Math, asked by subham90309, 3 months ago

the dimensions of a cuboid are in a ratio 3:2:1 and the total surface area is 1078cm^2 find the length of its on diagonal​

Answers

Answered by JaisonAFA
0

757cm i think it is correct answer

Answered by Dinosaurs1842
4

Given :-

  • The dimensions of a cuboid are in the ratio 3:2:1
  • Total surface area of the cuboid = 1078cm²

Aim :-

  • To find the length of it's diagonal.

Ratio :-

Ratio is the comparison of any two or more quantities represented in it's simplest form.

The dimensions of the cuboid :- 3:2:1 is thus in it's reduced form. Hence, let

The common factor by which these dimensions were cancelled out be x.

Therefore, the dimensions will be :- 3x, 2x and 1x respectively.

Total surface area :-

The total surface area of the cuboid is :-

 \longrightarrow \sf{2(lb \:  + bh \:  + lh)}

  • l represents length
  • b represents breadth
  • h represents height

Substituting the values,

 \implies \sf{1078} = 2[(3x)(2x) + (2x)(x) + (x)(3x)]

 \implies \sf{1078} = 2(6 {x}^{2}  + 2 {x}^{2}  + 3 {x}^{2} )

Adding the terms,

 \implies \sf{1078} = 2(11 {x}^{2})

Transposing 2 to the other side,

 \implies  \sf{\dfrac{1078}{2}  = 11 {x}^{2} }

Reducing to the lowest terms,

 \implies \sf{539} = 11 {x}^{2}

Transposing 11 to the other side,

 \implies \sf{ \dfrac{539}{11}  =  {x}^{2} }

Reducing to the lowest terms,

 \implies \sf{49} =  {x}^{2}

Transposing the power,

 \implies  \sf{\sqrt{49}  = x}

 \implies \sf{7} = x

Now that we have the value of x, the dimensions of the cuboid will be :-

  • 21 cm(3×7)
  • 14 cm (2×7)
  • 7 cm (1×7)

Verification :-

Let us verify the value of x.

 \implies \sf{1078} = 2[(21)(14) + (14)(7) + (7)(21)]

 \implies \sf{1078} = 2[294 + 98 + 147]

Adding,

 \implies \sf{1078} = 2[539]

LHS (Left hand side of the Equation) :-

➜ 1078

RHS (Right Hand Side of the Equation) :-

➜ 2(539)

➜ 1078

LHS and RHS are matching.

Hence verified.

Now that we have verified the values, let's find the length of the diagonal.

Diagonal :-

The diagonal of the cuboid :-

 \longrightarrow \sqrt{ \sf{(length)^{2}   + (breadth)^{2}  + (height)^{2} }}

Substituting the values,

 \implies \sqrt{ {21}^{2}  +  {14}^{2}  +  {7}^{2} }

 \implies \sqrt{441 + 196 + 49}

 \implies \sqrt{686}

Therefore the diagonal of the cuboid is 7√14 cm

Some more formulas :-

  • Total surface area of a cube = 6a²
  • The length of the diagonal = √3a

  • Total surface area of a cylinder = 2πrh + 2πr²
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