Question

The ratio of length, breadth and height of a rectangular solid is 3 : 2 : 1 and its total surface area is 352 sq. cm, then find the length.​

Answer 1

$12 \: cm$

Step-by-step explanation:

$given ,\: \: \: total \: \: surface \: \: area = 352cm^{2} \\ let, \: \: \: length \: (l) = 3x ,\: \: breadth \: (b) = 2x \: \: and \: \: heigth \: (h) = x \\ we \: \: know \: \: that \\ \boxed{total surface area of the cuboid = 2(lb +bh + hl)} \\ A/Q, \: \: \: 352 = 2(3x \times 2x + 2x \times x + x \times 3x) \\ 352 = 2(6 {x}^{2} + 2 {x}^{2} + 3 {x}^{2} ) \\ 352 = 2 \times 11 {x}^{2} \\ 352 = 22 {x}^{2} \\ {x}^{2} = 16 \\ x = \sqrt{16} = 4 \\putting \: \: value \: \: of \: x \\ so, \: \: \: length \: \: = 3x = 3 \times 4 = 12 \: cm$

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

The ratio of length, breadth and height of a rectangular solid is 3 : 2 : 1

Let assume that

Length of a rectangular solid be 3x cm

Breadth of a rectangular solid be 2x cm

Height of a rectangular solid be x cm

So, According to statement,

Total surface area of rectangular solid = 352 sq. cm

We know,

Total surface area of the Cuboid of length l, breadth b and height h is given by

$\boxed{\tt{ TSA_{(Cuboid)} \: = \: 2(lb \: + \: bh \: + \: hl) \: }}$

So, on substituting the values, we get

$\rm \: 352 = 2(3x \times 2x + 2x \times x + x \times 3x)$

$\rm \: 176 = {6x}^{2} + {2x}^{2} + {3x}^{2}$

$\rm \: 176 = 11 {x}^{2}$

$\rm \: {x}^{2} = 16$

$\rm\implies \:x = 4 \: \: \: \: \{ \: as\: x > \: 0 \: \}$

Hence,

Length of a rectangular solid = 3x = 3 × 4 = 12 cm

Breadth of a rectangular solid = 2x = 2 × 4 = 8 cm

Height of a rectangular solid = x = 4 cm

Thus,

Length of a rectangular solid be 12 cm.

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Additional Information

Formulas of Cube :-

Total Surface Area = 6(side)²

Curved Surface Area = 4(side)²

Volume of Cube = (side)³

Diagonal of a cube = √3(side)

Perimeter of cube = 12 x side

Formulas of Cuboid

Total Surface area = 2 (Length x Breadth + breadth x height + Length x height)

Curved Surface area = 2 height × (length + breadth)

Volume of the cuboid = length × breadth × height

Diagonal of the cuboid =√(l² + b² + h²)

Perimeter of cuboid = 4 (length + breadth + height)