Question

the Length breadth and height of a cuboid are in the ratio 3:4: 6 and its volume is 576cm2 .The whole surface area (in cm2) square of a cuboid is ​

Answer 1

Given :

• Length breadth and height of a cuboid are in the ratio 3 : 4 : 6.
• Its volume is 576 cm³.

To Find :

The total surface area cuboid.

Solution :

Analysis :

Here we have take a common ratio. Then using that we can find the dimensions of the cuboid. Then by using the formula for TSA we can find the area.

Required Formula :

• Volume = Length × Breadth × Height

• TSA = 2(lb + bh + lh)

where,

• l = Length
• b = Breadth
• h = Height

Explanation :

Let the common ratio be “x”.

• Length = 3x
• Breadth = 4x
• Height = 6x

We know that if we are given the ratio of length, breadth and height and the volume then our required formula is,

Volume = Length × Breadth × Height

where,

• Length = 3x
• Breadth = 4x
• Height = 6x
• Volume = 576 cm³

Using the required formula and substituting the required values,

⇒ Volume = Length × Breadth × Height

⇒ 576 = 3x × 4x × 6x

⇒ 576 = 72x³

⇒ 576/72 = x³

⇒ 8 = x³

⇒ ∛8 = x

⇒ ∛[2 × 2 × 2] = x

⇒ 2 = x

∴ x = 2.

The Dimensions :

1. Length = 3x = 3 × 2 = 6 cm
2. Breadth = 4x = 4 × 2 = 8 cm
3. Height = 6x = 6 × 2 = 12 cm

Area :

We know that if we are given the length, breadth and height and the total surface area is asked to find then our required formula is,

TSA = 2(lb + bh + lh)

where,

• l = 6 cm
• b = 8 cm
• h = 12 cm

Using the required formula and substituting the required values,

⇒ TSA = 2(lb + bh + lh)

⇒ TSA = 2[(6 × 8) + (8 × 12) + (6 × 12)]

⇒ TSA = 2[48 + 96 + 72]

⇒ TSA = 2[216]

⇒ TSA = 2 × 216

⇒ TSA = 432

∴ TSA = 432 cm².

The total surface area is 432 cm².

Answer 2

Step-by-step explanation:

★ Concept :-

Here the concept of Total Surface Area of Cuboid has been used. Through the given data we know that the dimensions of the cuboid are not given. So, by taking the common ratio for the given dimensions we will find out the dimensions. After that, using the formula of Total Surface Area of cuboid we can find the T.S.A of cuboid.

Let's do it !!!

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★ Formula Used :-

$\star\;\underline{\boxed{\bf{\purple{Volume\ of\ Cuboid\ =\ Length \times Breadth \times Height.}}}}$

$\star\;\underline{\boxed{\bf{\purple{T.S.A\ of\ Cuboid\ =\ 2(lb\ +\ bh\ +\ hl).}}}}$

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★ Solution :-

Given,

» Ratio of the dimensions of cuboid = 3 : 4 : 6.

» Volume of cuboid = 576cm³.

• Let the length, breadth, and height of the cuboid be 3x, 4x, and 6x.

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~ For the value of dimensions of cuboid ::

We know that,

$\sf \rightarrow {Volume\ of\ Cuboid\ =\ \bf Length \times Breadth \times Height}$

By applying the values, we get

$\sf \rightarrow {576\ =\ \bf 3x\ +\ 4x\ +\ 6x}$

$\sf \rightarrow {576\ =\ \bf 72x^3}$

$\sf \rightarrow {\cancel \dfrac{576}{72}\ =\ \bf x^3}$

$\sf \rightarrow {8\ =\ \bf x^3}$

$\sf \rightarrow {\sqrt[3]{8}\ =\ \bf x}$

$\sf \rightarrow {\sqrt[3]{(2 \times 2 \times 2)}\ =\ \bf x}$

$\sf \rightarrow {2\ =\ \bf x}$

$\bf \rightarrow {x\ =\ \blue {2.}}$

~ The dimensions :-

» Length = 3x = 3 × 2 = 6cm.

» Breadth = 4x = 4 × 2 = 8cm.

» Height = 6x = 6 × 2 = 12cm.

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~ For the value of total surface area of cuboid ::

We know that,

$\sf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \bf 2(lb\ +\ bh\ +\ hl)}$

By applying the values, we get

$\sf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \bf 2(6 \times 8\ +\ 8 \times 12\ +\ 12 \times 6)}$

$\sf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \bf 2(48\ +\ 96\ +\ 72)}$

$\sf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \bf 2(216)}$

$\sf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \bf 2 \times 216}$

$\bf : \implies {Total\ Surface\ Area\ of\ Cuboid\ =\ \orange {432cm^2.}}$

$\tt {\underline{\boxed{\tt T.S.A\ of\ Cuboid\ =\ {\purple{\bf 432cm^2.}}}}}$

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★ More to know :-

$\sf \leadsto {Total\ Surface\ Area\ of\ Cuboid\ =\ 2(lb\ +\ bh\ +\ hl).}$

$\sf \leadsto {Lateral\ Surface\ Area\ of\ Cuboid\ =\ 2h(l\ +\ b).}$

$\sf \leadsto {Volume\ of\ Cuboid\ =\ length \times breadth \times height.}$

$\sf \leadsto {Perimeter\ of\ Cuboid\ =\ 4(length\ +\ breadth\ +\ height).}$