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The length, width, and height of a rectangular solid are in the ratio of 3: 2: 1. If the volume of the box is 48 $cm 3$ then what is the total surface area of the box?

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c m 2

cm 2

cm 2

cm 2

answer is D.

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Detailed Solution

Given that, length, width, and height of a rectangular solid are in the ratio of 3: 2: 1. If the volume of the box is 48

cm 3

.
Let

x

be the common term of the ratio,

3 : 2 : 1

Then,
Length

= 3 x

Breadth

= 2 x

Height

= x

Now, we will calculate the volume of the box.
Formula that we will require to solve this question
Volume of cuboid

= l × b × h

where l is the length, b is the breadth and h is the height of the cuboid.

⇒ V = l b h ⇒ V = (3 x) (2 x) (x)

⇒ 48 = 6 x 3

⇒ x 3 = 8

⇒ x = 2

Then, to calculate the total surface area of the cuboid,
Total surface area of cuboid

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

Total surface area of the cuboid-

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

⇒ A = 2 (3 x ⋅ 2 x + 2 x ⋅ x + 3 x ⋅ x)

⇒ A = 22 x 2 ⇒ A = 22 (22) ⇒ A = 22 × 4

⇒ A = 88 c m 2

Thus, the total surface area of the box is

88 c m 2

.
Hence, option 4 is the correct option.

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