Question

the dimension of a cuboid are in ratio 2 ratio 3 ratio 4 and the total surface area is 5200 find the volume of the cuboid​

Answer 1

Given:

• Dimensions of cuboid = 2:3:4
• Total Surface Area = 5200 cm

To Find:

➜ Volume of cuboid = ?

Method:

Let

• Length = 2x
• Breadth = 3x
• Height = 4x

➳ TSA of cuboid = 2(lb + bh + hl)

➞ 5200 = 2(2x.3x + 3x.4x + 4x.2x)

➞ 5200 = 2(6x² + 12x² + 8x²)

➞ 5200 = 2(26x²)

➞ 5200 = 52x²

➞ x² = 5200 ÷ 52

➞ x² = 100

➞ x = √100

⇒ x = ± 10

Side cannot be negative.

∴ x = 10 cm

Now let's find the sides of the cuboid.

• Length = 2x = 2(10) = 20 cm
• Breadth = 3x = 3(10) = 30 cm
• Height = 4x = 4(10) = 40 cm

For finding Volume,

Volume = Length × Breadth × Height

➪ Volume = 20 × 30 × 40

➝ Volume = 24000 cm³

∴ Volume = 24000 cm³

Answer 2

Given :

• Dimensions of cuboid = 2:3:4
• Total Surface Area = 5200 cm²

To Find :

• Volume of cuboid = ?

Solution :

As, we are given dimensions = 2:3:4

Hence,

Let Length = 2x

Breadth = 3x

Height = 4x

Now, we know that

$\Large \underline{\boxed{\bf{ TSA \: of \: cuboid = 2(lb + bh + lh) }}}$

Now, by putting values,

$\sf : \implies 5200 = 2(2x \times 3x + 3x \times 4x + 2x \times 4x)$

$\sf : \implies 5200 = 2(6x^{2} + 12x^{2} + 8x^{2})$

$\sf : \implies 5200 = 2(26x^{2})$

$\sf : \implies 5200 = 2 \times 26x^{2}$

$\sf : \implies 5200 = 52x^{2}$

$\sf : \implies \dfrac{\cancel{5200}^{100}}{\cancel{52}} = x^{2}$

$\sf : \implies 100 = x^{2}$

$\sf : \implies \pm \sqrt{100} = x$

$\sf : \implies \pm \sqrt{(10)^{2}} = x$

$\sf : \implies \pm 10 = x$

As, we know that side cannot be negative.

$\sf : \implies x = 10 \: cm$

Hence,

• Length = 2x = 2 × 10 = 20 cm
• Breadth = 3x = 3 × 10 = 30 cm
• Height = 4x = 4 × 10 = 40 cm

Now, we know that,

$\Large \underline{\boxed{\bf{ Volume = Length \times Breadth \times Height }}}$

By putting values,

$\sf : \implies Volume = 20 \times 30 \times 40$

$\sf : \implies Volume = 24000$

Hence, Volume of cuboid is 24000 cm³.