Question

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​

Answer 1

757cm i think it is correct answer

Answer 2

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