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Mathematics

The length of the diagonal of a cuboid is 13213\sqrt{2} cm and its volume and total surface area is respectively 780 cm3 and 562 cm3. Find the dimension of the cuboid.

Mensuration

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Answer

Let the dimensions of the cuboid be l, b and h cm and diagonal be d cm.

Given,

Volume = 780 cm3

TSA = 562 cm2

d = 13213\sqrt{2}

By formula,

d2 = l2 + b2 + h2

(132)2(13\sqrt{2})^2 = l2 + b2 + h2

338 = l2 + b2 + h2……….(1)

We know that,

Volume of cuboid = l × b × h

780 = l × b × h ……….(2)

By formula,

TSA = 2(lb + bh + hl)

562 = 2(lb + bh + hl)……….(3)

By using identity,

(l + b + h)2 = l2 + b2 + h2 + 2(lb + bh + hl)

By substituting the values we get,

(l + b + h)2 = 338 + 562

(l + b + h)2 = 900

l + b + h = 900\sqrt{900}

l + b + h = 30……….(4)

From equation (2) & (4)

Sum of the dimensions = 30

Product of the dimensions = 780

780 = 5 x 12 x 13

and,

5 + 12 + 13 = 30.

∴ The dimensions are 5, 12 and 13 cm.

l = 5 cm, b = 12 cm, h = 13 cm.

Hence, dimensions of the cuboid = 5 cm, 12 cm and 13 cm.

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