Question

The breadth of a cuboid is one-fourth of length and height is two-third of the length. If the volume of the cuboid is 288 m', find the total surface area.​

Answer 1

Answer:

Total surface area of cuboid is 312 m².

Step-by-step-explanation:

Let the length, breadth and height of the cuboid be l, b and h respectively.

We have given that,

Breadth = ¼ × Length

⇒ b = ¼ × l - - ( 1 )

Also,

Height = ⅔ × Length

⇒ h = ⅔ × l - - ( 2 )

We know that,

Volume of cuboid = l × b × h - - [ Formula ]

⇒ 288 = l × ¼ × l × ⅔ × l - - [ From given, ( 1 ) & ( 2 ) ]

⇒ ( 288 × 4 × 3 ) ÷ ( 1 × 2 ) = l × l × l

⇒ 288 × 2 × 3 = l³

⇒ 144 × 2 × 2 × 3 = l³

⇒ l³ = 12 × 12 × 4 × 3

⇒ l³ = 12 × 12 × 12

⇒ l = 12 m. - - [ Taking cube roots ]

Now,

Breadth of cuboid = ¼ × Length

⇒ b = ¼ × 12

⇒ b = 12 ÷ 4

⇒ b = 3 m.

Now,

Height of cuboid = ⅔ × Length

⇒ h = ⅔ × 12

⇒ h = ( 2 × 12 ) ÷ 3

⇒ h = 24 ÷ 3

⇒ h = 8 m.

Now, we have,

• Length ( l ) = 12
• Breadth ( b ) = 3
• Height ( h ) = 8

Now, we know that,

Total surface area of cuboid = 2 ( lb + bh + lh )

⇒ TSAᶜᵘᵇᵒⁱᵈ = 2 [ ( 12 × 3 ) + ( 3 × 8 ) + ( 12 × 8 ) ]

⇒ TSAᶜᵘᵇᵒⁱᵈ = 2 [ 36 + 24 + 96 ]

⇒ TSAᶜᵘᵇᵒⁱᵈ = 2 [ 60 + 96 ]

⇒ TSAᶜᵘᵇᵒⁱᵈ = 2 × 156

⇒ TSAᶜᵘᵇᵒⁱᵈ = 312 m².

∴ Total surface area of cuboid is 312 m².

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Additional Information:

1. Cuboid:

A three dimensional figure having six rectangular faces is called cuboid.

2. It has total 8 vertices.

3. It has total 12 edges.

4. Volume of cuboid:

Length × Breadth × Height

5. Total surface area of cuboid:

2 ( lb + bh + lh )

6. Perimeter of cuboid:

4 ( l + b + h )

Answer 2

$\bf{\underline{Question:-}}$

The breadth of a cuboid is one-fourth of length and height is two-third of the length. If the volume of the cuboid is 288 m', find the total surface area.

$\bf{\underline{Given:-}}$

• Breadth = 1/4th of length
• height 2/3rd of length
• Volume of cuboid = 288m

$\bf{\underline{To\:Find:-}}$

• Total surface area = ?

$\bf{\underline{Solution:-}}$

Let ,

• the Lenght = x
• Breadth = y
• Height = z

Given = breadth = 1/4 of length

→ 1/4 × x ----eq(¡)

Similarly

→ 2/3 × x ----eq(¡¡)

$\bf{\underline{Formula:-}}$

Volume of cuboid = L × B × H

By eq(¡) & eq(¡¡)

→ 288 = X × ¼ × X × ⅔ × X

→$\sf \frac<strong> </strong>{288× 12}{2} = x^3$

→ 288×6 = x³

→ 1728 = x³

→ $\sf \sqrt[3]{1728}= x$

→ 12 = x

Calculating Breadth

→ b= 1/4 × x

→ b = 1/4 × 12

→ b = 1×12/4

→ b = 3cm

Same as height

→ h = 2/3 × x

→ h = 2/3 × 12

→ h = 2×4 = 8

→ h = 8cm

$\bf{\underline{Formula:-}}$

T.S.A = 2( L × B + B × H + H × L )

Calculating T.S.A of cuboid

T.S.A = 2 (12 × 3 + 3 × 8 + 8 × 12 )

T.S.A = 2 ( 36 + 24 + 96 )

T.S.A = 2 ( 156 )

T.S.A = 312 cm²

$\bf{\underline{Hence:-}}$

• The required total surface area of cuboid = 312 cm²