The length of longest pole that can be placed in the floor of a room is 12 m and the length of longest pole that can be placed in the room is 15 m. The height of the room is

Options

A. 3 M
B. 6 m
C. 9 m
D. 12 m

Answer

To find the height of the room given the lengths of the longest poles that can fit in the floor and the entire room, we can follow these steps:

### Step 1: Understand the Problem
We are given two lengths:
1. The longest pole that can be placed on the floor of the room is 12 m. This represents the diagonal of the floor.
2. The longest pole that can be placed in the room is 15 m. This represents the diagonal of the room including height.

### Step 2: Set Up the Equations
For the floor of the room, which we can assume to be rectangular, the diagonal \( d \) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
where \( l \) is the length and \( b \) is the breadth of the room. Since the diagonal is 12 m:
\[ 12^2 = l^2 + b^2 \]
This simplifies to:
\[ 144 = l^2 + b^2 \quad \text{(Equation 1)} \]

For the entire room, including height \( h \), the diagonal is given by:
\[ d = \sqrt{l^2 + b^2 + h^2} \]
Given that the diagonal is 15 m:
\[ 15^2 = l^2 + b^2 + h^2 \]
This simplifies to:
\[ 225 = l^2 + b^2 + h^2 \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2
From Equation 1, we know that \( l^2 + b^2 = 144 \). We can substitute this into Equation 2:
\[ 225 = 144 + h^2 \]

### Step 4: Solve for Height \( h \)
Now, we can isolate \( h^2 \):
\[ h^2 = 225 - 144 \]
\[ h^2 = 81 \]
Taking the square root of both sides gives:
\[ h = \sqrt{81} = 9 \text{ m} \]

### Conclusion
The height of the room is **9 meters**.

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