Question

The dimensions of a cuboid are in the ratio of 3:2:1. if it's total surface area is 1078cm², find it's volume

Answer 1

GivEn:-

• dimensions of a cuboid are in the ratio of 3:2:1.

• Total surface area of cuboid = 1078cm²

To find:-

• Volume of cuboid

SoluTion:-

Lets the dimensions of the cuboid be 3x, 2x and x.

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Reference of image is shown in diagram♡

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$\setlength{\unitlength}{0.68cm}\begin{picture}(12,4)\linethickness{0.3mm}\put(6,6){\line(1,0){5}}\put(6,9){\line(1,0){5}}\put(11,9){\line(0,-1){3}}\put(6,6){\line(0,1){3}}\put(4,7.3){\line(1,0){5}}\put(4,10.3){\line(1,0){5}}\put(9,10.3){\line(0,-1){3}}\put(4,7.3){\line(0,1){3}}\qbezier(6,6)(4,7.3)(4,7.3)\qbezier(6,9)(4,10.2)(4,10.3)\qbezier(11,9)(9.5,10)(9,10.3)\qbezier(11,6)(10,6.6)(9,7.3)\put(8,5.5){\sf{3x cm}}\put(4,6.3){\sf{2x cm}}\put(10,7.5){\sf{x cm}}\end{picture}$

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$\;\;\star\;{\underline{\underline{\sf{\pink{\;\;As\;per\;given\; Question\;:\;\:}}}}}$

• Total surface area of cuboid = 1078cm²

As we know that,

${\underline{\boxed{\sf{\purple{Total\;Surface\;Area\;of\;cuboid = 2(lb + bh + hl)}}}}}$

★ $\sf {\underline{\red{Put\;the\;givEn\;values:}}}$

$:\implies\sf 1078 = 2(3x \times 2x + 2x \times x + x \times 3x)\\\\ :\implies\sf 1078 = 2(6x^2 + 2x^2 + 3x^2)\\\\ :\implies\sf 1078 = 2(11x^2)\\\\ :\implies\sf 1078 = 22x^2\\\\ :\implies\sf x^2 = \cancel{ \dfrac{1078}{22}}\\\\ :\implies\sf x^2 = 49$

★ $\sf {\underline{\red{Taking\;sqrt\;both\;sides:}}}$

$:\implies\sf \sqrt{x^2} = \sqrt{49}\\\\:\implies{\underline{\boxed{\sf{\blue{x = 7\;cm}}}}}\;\bigstar$

Now,

Put the value of x in -

• Length of cuboid (3x) = 3 × 7 = 21cm

• Breadth of cuboid (2x) = 2 × 7 = 14cm

• Heigth of cuboid (x) = 7 = 7cm

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✇ Now, We have to find Volume of cuboid -

As we know that,

${\underline{\boxed{\sf{\purple{Volume\;of\;cuboid = l \times b \times h}}}}}$

★ $\sf {\underline{\red{Put\;the\;givEn\;values:}}}$

$:\implies\sf Volume = 21 \times 14 \times 7\\\\ :\implies{\underline{\boxed{\sf{\blue{Volume = 2058\;cm^3}}}}}\;\bigstar$

$\sf \underline{\therefore\;Volume\;of\;cuboid\;is\;2058\;cm^2}$

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Answer 2

Given :

Dimensions of a cuboid are in the ratio of 3:2:1

And total surface area of cuboid is 1078 cm²

To Find :

Volume of the cuboid

Solution :

Let the length ,breadth and height of the cuboid be 3x , 2x and x .

We know that

Surface area of a cuboid=2(lb+bh+hl)

$\sf1078=2(3x\times2x+2x\times\:x+x\times3x)$

$\sf1078=2(6x^2+2x^2+3x^2)$

$\sf1078=2\times11x^2$

$\sf\:x^2=\dfrac{1078}{22}$

$\sf\:x=\sqrt{49}$

$\sf\:x=7cm$

Thus, Dimensions of cubiod

• Length = 3x= 21cm
• Breadth = 2x = 14cm
• Height =x=7cm

We have to find the volume of cuboid

$\bf{Volume\:of\:cubiod=l\times\:b\times\:h}$

$\sf\:Volume=21\times14\times7$

$\sf\:Volume=147\times14$

$\sf\:Volume=2058cm^3$

Therefore, Volume of the cuboid is 2058 cm³.