Assume that aclass room is arectangular box where the floor is 8metres long and 6metres wide.If the distance from the floor to the ceiling(height of the room)is 3metres,find the diagonal distance between acorner of the room on the floor and the opposite corner on the ceiling.
√(8^2+6^2+3^2) = √(64+36+9) = √109
To find the diagonal distance between a corner of the room on the floor and the opposite corner on the ceiling, we can use the Pythagorean theorem.
Step 1: Find the diagonal distance on the floor.
Using the Pythagorean theorem, the diagonal distance on the floor (d1) can be found by:
d1 = √(length^2 + width^2)
Given:
Length = 8 meters
Width = 6 meters
Substituting the values into the formula:
d1 = √(8^2 + 6^2)
d1 = √(64 + 36)
d1 = √100
d1 = 10 meters
Step 2: Find the diagonal distance on the ceiling.
The diagonal distance on the ceiling will be the same as the diagonal distance on the floor.
d2 = d1 = 10 meters
Step 3: Find the height of the room.
Given:
Height = 3 meters
Step 4: Find the diagonal distance between a corner on the floor and the opposite corner on the ceiling.
To find the diagonal distance between the two corners, we can use the Pythagorean theorem again. This time, we will use the diagonal distance on the floor (d1), the diagonal distance on the ceiling (d2), and the height of the room.
Using the Pythagorean theorem:
diagonal distance = √(d1^2 + d2^2 + height^2)
Substituting the values:
diagonal distance = √(10^2 + 10^2 + 3^2)
diagonal distance = √(100 + 100 + 9)
diagonal distance = √209
diagonal distance ≈ 14.45 meters
Therefore, the diagonal distance between a corner of the room on the floor and the opposite corner on the ceiling is approximately 14.45 meters.
To find the diagonal distance between a corner on the floor and the opposite corner on the ceiling, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the diagonal distance is the hypotenuse of a right-angled triangle.
We can obtain the length of the diagonal distance using the following steps:
1. Calculate the length of the diagonal on the floor:
- Use the Pythagorean theorem to find the length of the diagonal on the floor.
- The length of the floor (8 meters) is one side of the right-angled triangle.
- The width of the floor (6 meters) is the other side of the right-angled triangle.
- Apply the Pythagorean theorem: diagonal^2 = length^2 + width^2.
- Substitute the values: diagonal^2 = 8^2 + 6^2.
2. Calculate the length of the diagonal on the ceiling:
- The length of the diagonal on the ceiling is equal to the length of the diagonal on the floor since the room is rectangular and the sides are parallel. Therefore, we can simply use the previous calculation.
3. Calculate the total diagonal distance:
- Add the diagonal distance on the floor and the diagonal distance on the ceiling to find the total diagonal distance.
Let's now calculate the diagonal distance:
Step 1: Calculate the length of the diagonal on the floor:
diagonal_floor^2 = 8^2 + 6^2
diagonal_floor^2 = 64 + 36
diagonal_floor^2 = 100
diagonal_floor = √100
diagonal_floor = 10 meters
Step 2: Calculate the length of the diagonal on the ceiling:
diagonal_ceiling = diagonal_floor = 10 meters
Step 3: Calculate the total diagonal distance:
total_diagonal_distance = diagonal_floor + diagonal_ceiling
total_diagonal_distance = 10 meters + 10 meters
total_diagonal_distance = 20 meters
Therefore, the diagonal distance between a corner of the room on the floor and the opposite corner on the ceiling is 20 meters.