The longest road that can be placed in a rectangular box = 20 cm and the sum of its length breadth and height is 30 cm. Find the total surface area of the box.

Given, the longest road that can be placed in a rectangular box, d = 20 cm

The longest rod which can be kept inside a rectangular box will be equal to the diagonal of the box.

By formula,

⇒ Diagonal² = l² + b² + h²

⇒ 20² = l² + b² + h²

⇒ 400 = l² + b² + h² ………………(1)

Given,

The sum of its length breadth and height equals to 30 cm.

⇒ l + b + h = 30 cm.

Squaring both sides, we get :

⇒ (l + b + h)² = 30²

⇒ l² + b² + h² + 2(lb + bh + hl) = 900

From equation (1), we get

⇒ 400 + 2(lb + bh + hl) = 900

⇒ 2(lb + bh + hl) = 900 - 400

⇒ 2(lb + bh + hl) = 500

Hence, the total surface area of the box = 500 cm².