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The dimensions of a cuboid are in the ratio 5: 3: 1 and its total surface area is $414 m 2$ . Find the dimensions.

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4m,3m,2m

15m,9m,3m

23m,3m,5m

34m,5m,6m

answer is B.

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Detailed Solution

It is given a cuboid of dimensions in the ratio 5:3:1 and its total surface area is

414 m 2

.
The surface area of a cuboid having a length

l

, breadth

b

and height

h

2 (l b + b h + h l)

.
Let the constant of proportionality be

x

.
The initial dimensions of the cuboid will be,

⇒ l = 5 x, b = 3 x, h = x

… (1)
Let the surface area of the cuboid be S.
The formula of the surface area of cuboid as,

S = 2 (l b + b h + h l)

… (2)
Substitute the values of l, b, h from equation (1) in equation (2) .

⇒ S = 2 (5 x × 3 x) + (3 x × x) + (x × 5 x)

⇒ S = 2 15 x 2 + 3 x 2 + 5 x 2

⇒ S = 2 × 23 x 2

⇒ S = 46 x 2

… (3)
Equate the given surface area with equation (3).
Since, the surface area of a cuboid is given to be 414

m 2

⇒ 46 x 2 = 414

⇒ 46 x 2 46 = 41446

⇒ x 2 = 9

⇒ x 2 = 32

⇒ x 2 = 32

⇒ x = 3

Substitute the value of

x = 3

in equation

1

.
Since,

l = 5 x

⇒ l = 5 × 3

Therefore l=15

m

Since,

b = 3 x

⇒ b = 3 × 3

Therefore b=9

m

Since,

h = x

⇒ h = 3

Therefore, h = 3m
Therefore, dimensions of a cuboid having surface area as

414 m 2

are 15m , 9m and 3 m.
Hence option (2) is correct.

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