The breadth of a room is twice its height and half its length and its volume is 1000 `m^3` Its dimensions are

Options

A. 20 m `xx` 10 m `xx`5 m
B. 10 m `xx`10 m `xx` 1 m
C. 40 m `xx` 5 m`xx`5 m
D. None of these

Answer

To find the dimensions of the room given the relationships between its height, breadth, and length, we can follow these steps:

### Step 1: Define the variables
Let:
- \( h \) = height of the room
- \( b \) = breadth of the room
- \( l \) = length of the room

### Step 2: Set up relationships based on the problem statement
According to the problem:
1. The breadth of the room is twice its height:
\[ b = 2h \]
2. The breadth of the room is half its length:
\[ b = \frac{1}{2}l \]

### Step 3: Express length in terms of height
From the equation \( b = 2h \), we can express \( l \) in terms of \( h \):
Substituting \( b \) into the second equation:
\[ 2h = \frac{1}{2}l \]

Multiplying both sides by 2 to eliminate the fraction:
\[ 4h = l \]

### Step 4: Write the volume equation
The volume \( V \) of the room is given by:
\[ V = l \times b \times h \]
Substituting the expressions for \( l \) and \( b \) in terms of \( h \):
\[ V = (4h) \times (2h) \times h \]
\[ V = 8h^3 \]

### Step 5: Set the volume equal to 1000 m³
We know the volume is 1000 m³:
\[ 8h^3 = 1000 \]

### Step 6: Solve for height \( h \)
To find \( h \), divide both sides by 8:
\[ h^3 = \frac{1000}{8} \]
\[ h^3 = 125 \]

Now, take the cube root of both sides:
\[ h = \sqrt[3]{125} \]
\[ h = 5 \, m \]

### Step 7: Find breadth \( b \) and length \( l \)
Now that we have \( h \), we can find \( b \) and \( l \):
1. Calculate breadth \( b \):
\[ b = 2h = 2 \times 5 = 10 \, m \]
2. Calculate length \( l \):
\[ l = 4h = 4 \times 5 = 20 \, m \]

### Conclusion
The dimensions of the room are:
- Height \( h = 5 \, m \)
- Breadth \( b = 10 \, m \)
- Length \( l = 20 \, m \)