Solving for Wall Breadth from Volume and Dimensions

This problem requires us to find the breadth of a wall given its total volume and relationships between its dimensions (breadth, height, and length). We need to use basic algebra and geometry principles to solve this.

Defining Wall Dimensions

Let's denote the dimensions of the wall as follows:

• Breadth = \( B \) meters
• Height = \( H \) meters
• Length = \( L \) meters

Relating Wall Dimensions

The question provides specific relationships between these dimensions:

• The wall is 5 times as high as it is broad: \( H = 5 \times B \)
• The wall is 8 times as long as it is high: \( L = 8 \times H \)

We can express the length \( L \) in terms of the breadth \( B \) by substituting the first relation into the second:

\( L = 8 \times (5 \times B) \) \( L = 40 \times B \)

So, we have the dimensions in terms of \( B \):

• Breadth = \( B \)
• Height = \( 5B \)
• Length = \( 40B \)

Calculating Wall Volume

The volume of a rectangular object (like a wall) is given by the formula:

Substituting the expressions for \( L \), \( B \), and \( H \) in terms of \( B \):

Solving for Breadth (B)

We are given that the volume \( V \) is 12.8 cubic meters. We can set up the equation:

Now, we solve for \( B \):

1. Divide both sides by 200: \( B^3 = \frac{12.8}{200} \)
2. Simplify the fraction: \( B^3 = \frac{128}{2000} \) \( B^3 = \frac{16}{250} \) \( B^3 = \frac{8}{125} \)
3. Take the cube root of both sides: \( B = \sqrt[3]{\frac{8}{125}} \) \( B = \frac{\sqrt[3]{8}}{\sqrt[3]{125}} \) \( B = \frac{2}{5} \, \text{meters} \)
4. Convert the fraction to a decimal: \( B = 0.4 \, \text{meters} \)

Converting Units to Centimeters

The options are given in meters and centimeters. The calculated breadth is 0.4 meters. To convert this to centimeters, we multiply by 100 (since 1 meter = 100 centimeters):

Conclusion

The breadth of the wall is calculated to be 0.4 meters, which is equal to 40 centimeters.