The breadth ‘b’ of a room is twice its height and half of its length. Find the length of the longest diagonal of the room.

ATQ, Given information: The breadth 'b' of the room is twice its height. The breadth 'b' is half of its length. Let's assign variables to the dimensions of the room: Height: h, Breadth: b, Length: l From the given information, we can write the following equations: Equation 1: b = 2h Equation 2: b = l/2 We can solve these equations to find the values of 'b' and 'l' in terms of 'h'. From Equation 1, we have: b = 2h Substituting this value of 'b' into Equation 2, we get: 2h = l/2 Multiplying both sides by 2, we get: 4h = l Now we have expressions for 'b' and 'l' in terms of 'h': b = 2h l = 4h To find the length of the longest diagonal of the room, we need to find the diagonal of the rectangular room, which can be calculated using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. In this case, the longest diagonal of the room will be the hypotenuse of a right-angled triangle with sides of length 'b' and 'l'. Using the Pythagorean theorem, we can calculate the length of the diagonal (D) as follows: D² = b² + l² Substituting the values of 'b' and 'l', we get: D² = (2h)² + (4h)² D² = 4h² + 16h² D² = 20h² Taking the square root of both sides, we get: D = √(20h²) D = 2√5 × h Therefore, the length of the longest diagonal of the room is  times the height of the room.