Question

Q.13.If the length,breadth and height of a cuboid are in the ratio5:3:2and its total

surface area is1550sq.m. Find:(i)the dimensions of the cuboid (ii)volume of the cuboid.​

Answer 1

Question:-

If the length,breadth and height of a cuboid are in the ratio5:3:2and its total surface area is1550sq.m. Find:(i)the dimensions of the cuboid (ii)volume of the cuboid.

Answer:-

• The length, Breadth and height of cuboid is 25 m,15 m and 10 m.
• The volume of cuboid is 3750 m³.

To find:-

• the dimensions of the cuboid
• volume of the cuboid.

Solution:-

• Ratio = 5:3:2

Put x in the ratio,

• Length = 5x
• Breadth = 3x
• Height = 2x
• Total surface area = 1550 m²

(i)

$\large { \mathfrak{ \green{ \underline{ \purple{ sa = 2(lb + bh + lh)}}}}}$

where,

• l = length
• b = breadth
• h = height

According to question,

$\large{ \tt: \implies \: \: \: \: \: \: \: 2(lb + bh + lh) = 1550}$

$\large{ \tt : \implies \: \: \: \: \: 2(5x \times 3x + 3x \times 2x + 5x \times 2x) = 1550}$

$\large{ \tt : \implies \: \: \: \: \: 2(31 {x}^{2} ) = 1550}$

$\large{ \tt : \implies \: \: \: \: \: 31 {x}^{2} = \frac{1550}{2} }$

$\large{ \tt : \implies \: \: \: \: \: {x}^{2} = \frac{775}{31} }$

$\large{ \tt : \implies \: \: \: \: \: x = \sqrt{25} }$

$\large{ \tt : \implies \: \: \: \: \: x = 5 \: m}$

• The value of x is 5 m
• Length = 5x = 5×5 = 25 m
• Breadth = 3x = 3×5 = 15 m
• Height = 2x = 2×5 = 10 m

The length, Breadth and height of cuboid is 25 m,15 m and 10 m.

(ii)

$\large { \mathfrak{ \green{ \underline{ \purple{ volume = l \times b \times h}}}}}$

According to question,

$\large{ \tt : \implies \: \: \: \: volume = 25 \times 15 \times 10}$

$\large{ \tt : \implies \: \: \: \: volume = 3750 \: {m}^{3} }$

• The volume of cuboid is 3750 m³.

Answer 2

Given

• The length, breadth and height of a Cuboid are in the ratio 5:3:2.
• Total surface area of Cuboid is 1550 m².

⠀

To find

• (i) The dimensions of the Cuboid.
• (ii) Volume of the Cuboid.

⠀

Solution

• Let the ratio be x.

Then,

⠀⠀❍ Length = 5x

⠀⠀❍ Breadth = 3x

⠀⠀❍ Height = 2x

⠀

$\boxed{\sf{\orange{(i)}}}$

• As the CSA of cuboid is given in the question, we will use the formula

⠀

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

⠀

Here,

• l = 5x
• b = 3x
• h = 2x

⠀

★ Putting the values

$\tt\longmapsto{1550 = 2\bigg\lgroup{(5x)(3x) + (3x)(2x) + (2x)(5x)}\bigg\rgroup}$

⠀

$\tt\longmapsto{1550 = 2\bigg\lgroup{15x^2 + 6x^2 + 10x^2}\bigg\rgroup}$

⠀

$\tt\longmapsto{1550 = 2\bigg\lgroup{31x^2}\bigg\rgroup}$

⠀

$\tt\longmapsto{31x^2 = \dfrac{1550}{2}}$

⠀

$\tt\longmapsto{31x^2 = 775}$

⠀

$\tt\longmapsto{x^2 = \dfrac{775}{31}}$

⠀

$\tt\longmapsto{x^2 = 25}$

⠀

$\tt\longmapsto{x = \sqrt{25}}$

⠀

$\bf\longmapsto{x = 5}$

⠀

$\large{\boxed{\boxed{\sf{Dimensions\: of\: Cuboid :-}}}}$

⠀⠀⠀❍ Length = 5x = 25m

⠀⠀⠀❍ Breadth = 3x = 15m

⠀⠀⠀❍ Height = 2x = 10m

⠀

$\boxed{\sf{\orange{(ii)}}}$

• Finding Volume of the Cuboid.

⠀

$\: \: \: \: \: \: \: \: \: \: \: \: \boxed{\tt{\bigstar{Volume_{(Cuboid)} = l \times b \times h{\bigstar}}}}$

⠀

$\tt:\implies\: \: \: \: \: \: \: \: {Volume = 25 \times 15 \times 10}$

⠀

$\bf:\implies\: \: \: \: \: \: \: \: {Volume = 3,750}$

⠀

Hence,

• The volume of the cuboid is 3,750 m³.

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