Question

the length breadth and height are in ratio 3:1:2. the surface area of cuboid is 1408cm find length breadth and height ​

Answer 1

• Length of cuboid is 24 cm.
• Breadth of cuboid is 8 cm.
• Height of cuboid is 16 cm.

Step-by-step explanation:

Given:-

• Length, Breadth and Height of cuboid are in ratio of 3:1:2.
• Surface area of cuboid is 1408 cm².

To find:-

• Length of cuboid.
• Breadth of cuboid.
• Height of cuboid.

Solution:-

Let, Length of cuboid be 3x.

Breadth of cuboid be 1x or x.

And Height of cuboid be 2x.

Surface area of cuboid = 2(lb + bh + lh)

• Here, l, b and h are length, breadth and height of cuboid.

Putting l, b ,h and surface area in formula:

$\longrightarrow \sf 1408 = 2 \times [(3x \times x) + (x \times 2x)+(2x \times 3x)]$

$\longrightarrow \sf 1408 = 2 \times (3x^{2} - 2x^{2} + 6x^{2})$

$\longrightarrow \sf 1408 = 6x^{2} + 4x^{2} + 12x^{2}$

$\longrightarrow \sf 1408 = 22x^{2}$

$\longrightarrow \sf \cfrac{1408}{22} = x^{2}$

$\longrightarrow \sf 64 = x^{2}$

$\longrightarrow \sf \sqrt{64} = x$

$\longrightarrow \sf 8 = x$

Or, x = 8

Verification:-

$\longrightarrow \sf 1408 = 2 \times [(3x \times x) + (x \times 2x)+(2x \times 3x)]$

$\longrightarrow \sf 1408 = 2 \times (3x^{2} - 2x^{2} + 6x^{2})$

• Put x = 8

$\longrightarrow \sf 1408 = 2 \times (3 \times (8)^{2} + 2 \times (8)^{2} + 6 \times (8)^{2}$

$\longrightarrow \sf 1408 = 2 \times (3 \times 64 + 2 \times 64 + 6 \times 64$

$\longrightarrow \sf 1408 = 2 \times (192 + 128 + 384)$

$\longrightarrow \sf 1408 = 384 + 256 + 768$

$\longrightarrow \sf 1408 = 1408$

Hence, Verified!!

Length = 3x = 3×8 = 24 cm.

Breadth = x = 8 cm.

Height = 2x = 2×8 = 16 cm.

Answer 2

Answer:

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: Question- \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

The length, breadth and height are in ratio 3:1:2.

The surface area of cuboid is 1408 cm².

Find length ,breadth and height .

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: concept- \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

As per provided in the given Question ,

The length,breadth and height of a cuboid are in ratio 3:1:2.

The surface area of cuboid is given 1408 cm².

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: To\: find- \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

• Length (l)
• Breadth (b)
• Height (h)

$\huge \underbrace{ }$

Of a Cuboid

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: Formula - \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

$\large \mid{ \underbrace{\overbrace \bold \color{lime}{2(lb + bh + lh)}}} \mid$

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: Let- \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

• The length of the cuboid be = 3x
• The breadth of the cuboid be = x
• The height of the cuboid be = 2x

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: Solution- \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

Let us use the formula,

$\large \rm \red2(lb + bh + hl) \\ </p><p>\rm\green (as \: the \: given$

$\rm \green{surface \: area \: is \: 1408 )\: so}$

$\large\rm \purple{2(lb + bh + hl) = 1408} \\ \large \rm\bold { = > 2(3x \times x + x \times 2x + 3x \times 2x )= 1408} \\ \large \rm\bold{ = > 2(3 {x}^{2} + 2 {x}^{2} + 6 {x}^{2} = 1408 } \\ \large \rm\bold{ = > 6x {}^{2} + 4 {x}^{2} + 12 {x}^{2} = 1408 }\\ \large \rm \bold{ = > 22 {x}^{2} = 1408 }$

$\large \rm \bold{ = > {x}^{2} = \cancel\frac{1408}{22} } \\ \large \rm\bold {= > x {}^{2} = 64} \\ \large \rm \bold{ = > x = \sqrt{64} } \\ \large \rm \bold{ = > x = 8}$

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: Therefore \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

$\large \rm \pink{length = 3x} \\ = 3 \times 8 \\ = 24cm \\ \\ \large \rm \blue{breadth = x }\\ = 8cm \\ \\ \large \rm \color{lime}{ height = 2x} \\ 2 \times 8 \\ 16cm$

So ,

• the length is 24cm
• the breadth is 8 cm
• the height is 16cm

$\large \bigstar \bf {\underbrace{ \color{navy}{ \: \: \: \: \: \: \: \: \: \: additional\: information \: \: \: \: \: \: \: \: \: \: \: }} }\bigstar$

• The Cuboid is a three-dimensional figure
• All are sides aren't equal
• All the 6 faces are rectangle
• The faces of the cuboid are parallel.

A cuboid has

• 12 edges
• 6 faces
• 8 vertices.