Understanding the Wall Volume Problem

This question asks us to find the breadth of a wall given its volume and the relationships between its dimensions (breadth, height, and length). The volume is provided in cubic meters, while the options for breadth are in centimeters, so we need to be careful with unit conversions.

Setting up the Dimensions

Let's define the dimensions of the wall using a single variable. Let the breadth of the wall be \(b\).

• The height is 5 times as high as it is broad. So, Height (\(h\)) = \(5 \times b = 5b\).
• The length is 8 times as long as it is high. So, Length (\(l\)) = \(8 \times h\).

Now, substitute the expression for height (\(h = 5b\)) into the equation for length:

Length (\(l\)) = \(8 \times (5b) = 40b\).

So, the dimensions in terms of breadth \(b\) are:

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

Calculating the Volume

The volume of a rectangular wall (which is a rectangular prism) is given by the formula:

Volume (\(V\)) = Length \(\times\) Breadth \(\times\) Height

Substitute the expressions for length, breadth, and height in terms of \(b\) into the volume formula:

\(V = (40b) \times (b) \times (5b)\)

\(V = 40 \times 1 \times 5 \times b \times b \times b\)

\(V = 200 b^3\)

Using the Given Volume to Find Breadth

We are given that the volume of the wall is \(12.8 \text{ m}^3\).

So, we have the equation:

\(200 b^3 = 12.8 \text{ m}^3\)

Now, we need to solve for \(b\). Divide both sides by 200:

\(b^3 = \frac{12.8}{200}\)

To make the division easier, we can write 12.8 as 128/10 and 200 as 2000/10:

\(b^3 = \frac{128/10}{200} = \frac{128}{10 \times 200} = \frac{128}{2000}\)

Simplify the fraction by dividing the numerator and denominator by common factors. Divide both by 8:

\(b^3 = \frac{128 \div 8}{2000 \div 8} = \frac{16}{250}\)

Divide both by 2:

\(b^3 = \frac{16 \div 2}{250 \div 2} = \frac{8}{125}\)

So, \(b^3 = \frac{8}{125}\). To find \(b\), take the cube root of both sides:

\(b = \sqrt[3]{\frac{8}{125}}\)

\(b = \frac{\sqrt[3]{8}}{\sqrt[3]{125}}\)

We know that \(2^3 = 8\) and \(5^3 = 125\), so \(\sqrt[3]{8} = 2\) and \(\sqrt[3]{125} = 5\).

\(b = \frac{2}{5}\)

In decimal form, \(b = 0.4\).

Since the volume was in cubic meters (\(\text{m}^3\)), the breadth \(b\) is in meters. So, \(b = 0.4 \text{ meters}\).

Converting Breadth to Centimeters

The options are given in centimeters, so we need to convert the breadth from meters to centimeters. We know that:

\(1 \text{ meter} = 100 \text{ centimeters}\)

So, to convert 0.4 meters to centimeters, multiply by 100:

Breadth in cm = \(0.4 \times 100 \text{ cm}\)

Breadth in cm = \(40 \text{ cm}\)

The breadth of the wall is 40 cm.

Summary of Steps

1. Define dimensions using a variable (breadth \(b\)).
2. Express height and length in terms of \(b\).
3. Write the volume formula for a rectangular prism.
4. Substitute the expressions for dimensions into the volume formula.
5. Set the volume expression equal to the given volume (12.8 m³).
6. Solve the resulting equation for \(b\).
7. Convert the value of \(b\) from meters to centimeters.

Revision Table: Key Concepts

Additional Information: Understanding Volume and Dimensions

Volume is a measure of three-dimensional space. For simple shapes like a wall (which we model as a rectangular prism), it's calculated by multiplying the three dimensions: length, breadth (or width), and height. Ensuring all dimensions are in the same unit is crucial before calculating volume. If the volume is given in cubic meters (\(\text{m}^3\)), any linear dimension calculated directly from the volume formula will be in meters. If the volume is given in cubic centimeters (\(\text{cm}^3\)), the linear dimension will be in centimeters.

In this problem, we used algebraic substitution to express all dimensions in terms of a single unknown variable, the breadth. This is a common technique in geometry and measurement problems where relationships between different quantities are given. Solving the resulting equation involving a cubic term (\(b^3\)) required finding the cube root. Recognizing perfect cubes like 8 (\(2^3\)) and 125 (\(5^3\)) helps in simplifying cube roots of fractions.