Question

The sum of the length, breadth and height of a cuboid is `5sqrt(3)` cm and length of its diagonal is `3sqrt(5)` cm, then its total surface area is ______.

Answer 1

The sum of the length, breadth and height of a cuboid is `5sqrt(3)` cm and length of its diagonal is `3sqrt(5)` cm, then its total surface area is 30 cm².

Explanation:

Given: `l + b + h = 5sqrt(3)` cm 

Diagonal `d = sqrt(l^2 + b^2 + h^2) = 3sqrt(5)` cm

(Use: d² = l² + b² + h² and Total surface area = 2(lb + bh + lh).)

Step wise calculation:

1. Compute squares:

`(l + b + h)^2 = (5sqrt(3))^2 = 75`

`d^2 = (3sqrt(5))^2 = 45`

So l² + b² + h² = 45.

2. Expand (l + b + h)²: 

l² + b² + h² + 2(lb + bh + lh) = 75

3. Substitute l² + b² + h² = 45:

45 + 2(lb + bh + lh) = 75

⇒ 2(lb + bh + lh) = 30

⇒ (lb + bh + lh) = 15

4. Total surface area = 2(lb + bh + lh)

= 2 × 15

= 30 cm²