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The length, the breadth and the height of a cuboid are in the ratio of 5:3:2. If its volume is $240 {cm}^{3}$ then find the total surface area of the cuboid in ${cm}^{3} .$

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248 {cm}^{3}

228 {cm}^{3}

238 {cm}^{3}

268 {cm}^{3}

answer is A.

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Detailed Solution

The ratio of a cuboid's length, breadth, and height=5:3:2
A variable x that allows us to express the ratios as proportions 5x, 3x, and 2x.
The volume of the cuboid =

lbh

where

l = 5 x,

b = 3 x

h = 2 x

Therefore, substituting the values,
Volume

= (5 x) (3 x) (2 x)

Volume = 30 x^{3}

Now, according to the question, the volume of a cuboid

= 240 {cm}^{3}

So, we can equate obtained volume to given volume as,

30 x^{3} = 240 {cm}^{3}

\Rightarrow x^{3} = 8 {cm}^{3}

\Rightarrow x = 2 cm

So, now we can obtain the length of cuboid as

= 5 \times 2 = 10 cm

Breadth of cuboid

= 3 \times 2 = 6 cm

and
Height of the cuboid

= 2 \times 2 = 4 cm

The total surface area of cuboid,

\Rightarrow 2 (lb + bh + hl)

\Rightarrow 2 (10 \times 6 + 6 \times 4 + 4 \times 10)

\Rightarrow 2 (60 + 24 + 40)

\Rightarrow 2 (124)

= 248 {cm}^{3}

So, the total surface area of a cuboid is

= 248 {cm}^{3}

Correct option is 1.

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