The length of longest pole that can be placed on the floor of a room is 10 m and the length of the longest pole that can be placed in the room is `10sqrt2` m. The height of the room is ---

Options

A. 6 m
B. 7.5 m
C. 8 m
D. 10 m

Answer

To find the height of the room based on the given information, we can follow these steps:

### Step 1: Understand the Problem
We know that the longest pole that can be placed on the floor of the room is 10 m. This means that the diagonal of the rectangular base (length and breadth) of the room is 10 m.

### Step 2: Use the Pythagorean Theorem
Let the length of the room be \( L \) and the breadth be \( B \). According to the Pythagorean theorem, the diagonal \( D \) of the rectangle formed by the length and breadth can be expressed as:
\[ D = \sqrt{L^2 + B^2} \]
Since the diagonal is 10 m, we can write:
\[ \sqrt{L^2 + B^2} = 10 \]
Squaring both sides gives:
\[ L^2 + B^2 = 100 \quad \text{(Equation 1)} \]

### Step 3: Consider the Diagonal of the Room
The longest pole that can be placed in the room (which includes the height) is given as \( 10\sqrt{2} \) m. The diagonal \( D' \) of the room can be expressed as:
\[ D' = \sqrt{L^2 + B^2 + H^2} \]
Substituting the value of \( D' \):
\[ \sqrt{L^2 + B^2 + H^2} = 10\sqrt{2} \]
Squaring both sides gives:
\[ L^2 + B^2 + H^2 = 200 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we know \( L^2 + B^2 = 100 \). We can substitute this into Equation 2:
\[ 100 + H^2 = 200 \]

### Step 5: Solve for Height \( H \)
Now, we can solve for \( H^2 \):
\[ H^2 = 200 - 100 \]
\[ H^2 = 100 \]
Taking the square root of both sides gives:
\[ H = 10 \text{ m} \]

### Final Answer
The height of the room is **10 m**.

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