Calculating the Length of a Cylindrical Lead Wire

This problem involves the concept of volume conservation. When a material like lead is melted and drawn into a different shape, its total volume remains constant. We are given the volume of lead in cubic decimetres (dm³) and the diameter of the cylindrical wire it is drawn into in centimetres (cm). We need to find the length of the wire in metres (m).

Understanding the Given Information

• Volume of lead = 3.96 cubic dm ($ \text{dm}^3 $)
• The lead is drawn into a cylindrical wire.
• Diameter of the cylindrical wire = 0.6 cm
• We need to find the length of the wire in metres (m).

Step-by-Step Solution

Step 1: Convert Units

First, we need to work with consistent units. The diameter is given in centimetres (cm), and the volume is in cubic decimetres (dm³). Let's convert the volume from cubic decimetres to cubic centimetres (cm³). We know that 1 decimetre (dm) is equal to 10 centimetres (cm).

So, 1 cubic decimetre ($1 \text{ dm}^3$) is equal to $(10 \text{ cm})^3$.

$1 \text{ dm}^3 = 10 \text{ cm} \times 10 \text{ cm} \times 10 \text{ cm} = 1000 \text{ cm}^3$

Now, convert the given volume of lead:

Volume of lead = $3.96 \text{ dm}^3 = 3.96 \times 1000 \text{ cm}^3 = 3960 \text{ cm}^3$

Step 2: Determine the Radius of the Wire

The diameter of the cylindrical wire is 0.6 cm. The radius ($r$) is half of the diameter.

$r = \frac{\text{Diameter}}{2} = \frac{0.6 \text{ cm}}{2} = 0.3 \text{ cm}$

Step 3: Use the Volume Formula for a Cylinder

The volume of a cylinder is given by the formula:

$V = \pi r^2 L$

Where $V$ is the volume, $r$ is the radius, and $L$ is the length (or height) of the cylinder.

In this problem, the volume of the cylindrical wire is equal to the volume of the lead used, which is 3960 cm³. We know the radius is 0.3 cm. We need to find the length $L$ in centimetres first.

Substitute the known values into the formula:

$3960 \text{ cm}^3 = \pi (0.3 \text{ cm})^2 L$

$3960 = \pi (0.09) L$

Step 4: Solve for the Length (L)

We can use the value of $\pi \approx \frac{22}{7}$ for calculation.

$3960 = \frac{22}{7} \times 0.09 \times L$

To find $L$, rearrange the equation:

$L = \frac{3960 \times 7}{22 \times 0.09}$

To simplify the calculation, multiply the numerator and denominator by 100 to remove the decimal from 0.09:

$L = \frac{3960 \times 7 \times 100}{22 \times 0.09 \times 100} = \frac{3960 \times 7 \times 100}{22 \times 9}$

Now, perform the divisions:

$3960 \div 9 = 440$

$440 \div 22 = 20$

So, $L = 20 \times 7 \times 100$

$L = 140 \times 100$

$L = 14000 \text{ cm}$

Step 5: Convert Length to Metres

The length is currently in centimetres. We need to convert it to metres. We know that 1 metre (m) = 100 centimetres (cm).

To convert centimetres to metres, divide by 100:

$L \text{ in metres} = \frac{14000 \text{ cm}}{100 \text{ cm/m}}$

$L \text{ in metres} = 140 \text{ m}$

Thus, the length of the wire is 140 metres.

Summary of Calculations

Final Answer

The length of the cylindrical wire is 140 metres.

Revision Table: Volume and Unit Conversions

Additional Information: Applications of Volume Calculation

Calculating volumes is essential in various real-world applications, especially in manufacturing and engineering. For example:

• Material Estimation: Calculating how much material (like metal, plastic, or liquid) is needed to create an object of a specific shape and size. This helps in estimating costs and planning production.
• Capacity Measurement: Determining the capacity of containers, tanks, or pipes.
• Density Calculation: Using volume and mass to find the density of a substance, which is a key physical property.
• Civil Engineering: Calculating the volume of concrete, soil, or other materials needed for construction projects like roads, bridges, and buildings.
• Fluid Dynamics: Understanding the flow rate of liquids or gases through pipes, which depends on the volume of the pipe and the speed of the fluid.

Problems like converting the volume of lead into a wire length are practical examples of how geometry and volume calculations are used in material science and manufacturing processes like wire drawing.