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Question
The sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm and the length of its diagonal is `2sqrt(3)` cm. The total surface area of the cuboid is ______.
Options
48 cm2
72 cm2
96 cm2
108 cm2
Solution
The sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm and the length of its diagonal is `2sqrt(3)` cm. The total surface area of the cuboid is `underline(bb(96 cm^2)`.
Explanation:
Let Length = l
Breadth = b
Height = h
Given that
Sum of the length, breadth and height of a cuboid is `6sqrt(3)` cm
l + b + h = `6sqrt(3)` cm ......(1)
Also, given that
Length of its diagonal is `2sqrt(3)` cm
`sqrt(l^2 + b^2 + h^2) = 2sqrt(3)` ......(2)
We need to find
Total surface area of the cuboid
Now, Total surface area of cuboid = 2(lb + bh + lh)
From (1)
l + b + h = `6sqrt(3)` cm
Squaring both sides
(l + b + h)2 = `(6sqrt(3))^2`
l2 + b2 + h2 + 2lb + 2bh + 2lh = `6^2 xx (sqrt(3))^2`
l2 + b2 + h2 + 2lb + 2bh + 2lh = 36 × 3
l2 + b2 + h2 + 2lb + 2bh + 2lh = 108
From (1)
`sqrt(l^2 + b^2 + h^2) = 2sqrt(3)`
Squaring both sides
`(sqrt(l^2 + b^2 + h^2))^2 = (2sqrt(3))^2`
l2 + b2 + h2 = 22 × `(sqrt(3))^2`
l2 + b2 + h2 = 4 × 3
l2 + b2 + h2 = 12
12 + 2lb + 2bh + 2lh = 108
2lb + 2bh + 2lh = 108 – 12
2lb + 2bh + 2lh = 96
Total surface area = 96 cm2
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