The volume of a wall, 5 times as high as its breadth and 8 times as long as it is high is `18225" m"^(3)`. Find the breadth of the wall.

Options

A. `32.5` m
B. 5 m
C. `4.5` m
D. `3.5` m

Answer

To solve the problem, we need to find the breadth of the wall given its volume and the relationships between its dimensions. Let's denote:

- Breadth of the wall = \( b \)
- Height of the wall = \( h \)
- Length of the wall = \( l \)

According to the problem:

1. The height of the wall is 5 times its breadth:
\[ h = 5b \]

2. The length of the wall is 8 times its height:
\[ l = 8h \]

3. The volume of the wall is given by the formula:
\[ \text{Volume} = l \times b \times h \]
We know the volume is \( 18225 \, m^3 \).

Now, we can substitute the expressions for \( h \) and \( l \) in terms of \( b \) into the volume formula.

### Step 1: Substitute \( h \) and \( l \) into the volume formula
Substituting \( h = 5b \) and \( l = 8h = 8(5b) = 40b \):
\[ \text{Volume} = l \times b \times h = (40b) \times b \times (5b) \]

### Step 2: Simplify the expression
Now, simplify the expression:
\[ \text{Volume} = 40b \times b \times 5b = 200b^3 \]

### Step 3: Set the volume equal to the given volume
Now, we set the expression equal to the given volume:
\[ 200b^3 = 18225 \]

### Step 4: Solve for \( b^3 \)
To find \( b^3 \), divide both sides by 200:
\[ b^3 = \frac{18225}{200} \]
Calculating the right side:
\[ b^3 = 91.125 \]

### Step 5: Take the cube root to find \( b \)
Now, take the cube root of both sides to find \( b \):
\[ b = \sqrt[3]{91.125} \]

Calculating this gives:
\[ b \approx 4.48 \, m \]

### Conclusion
Thus, the breadth of the wall is approximately \( 4.48 \, m \).