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Question

A cuboid has a total surface area of 96 cm2. The sum of the squares of its length, breadth and height (in cm) is 48. The height of the cuboid is ________.

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Solution


Let the length, breadth and height of the cuboid be l cm, b cm and h cm, respectively.

Total surface area of the cuboid = 96 cm2 (Given)

⇒ 2(lb + bh + hl) = 96 cm2 .....(1)

Also,

l2 + b2 + h2 = 48 cm2 (Given)

⇒ 2(l2 + b2 + h2) = 2 × 48 = 96 cm2 .....(2)

Subtracting (1) from (2), we get

2(l2 + b2 + h2) − 2(lb + bh + hl) = 96 cm2 − 96 cm2 = 0

⇒ (l2 − 2lb + b2) + (b2 − 2bh + h2) + (h2 − 2hl + l2) = 0

⇒ (l − b)2 + (b − h)2 + (h − l)2 = 0

⇒ l − b = 0, b − h = 0, h − l = 0

⇒ l = b, b = h, h = l

⇒ l = b = h .....(3)

From (2) and (3), we get

h2 + h2 + h2 = 48 cm2

⇒ 3h2 = 48

⇒ h2 = 16

⇒ h = 4 cm

Thus, the height of the cuboid is 4 cm.

A cuboid has a total surface area of 96 cm2. The sum of the squares of its length, breadth and height (in cm) is 48. The height of the cuboid is ___4 cm___.

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