Calculating Metal Volume in a Cylindrical Tube

This problem asks us to find the volume of metal used to create a hollow cylindrical tube. The tube is open at both ends and has a specific thickness. We are given the outer radius, the thickness, and the length. To find the volume of the metal, we need to calculate the volume of the outer cylinder (including the empty space) and subtract the volume of the inner empty space.

Understanding the Key Information

Let's list the given values:

• Thickness of the metal sheet (t) = 0.5 cm
• Outer radius of the tube (R) = 4 cm
• Length of the tube (L) = 2 m

The tube is open at both ends, which means we don't need to consider end caps when calculating the volume of the metal part of the cylinder walls.

Units Conversion

Before performing any calculations, it's crucial to ensure all dimensions are in the same units. The thickness and radius are in centimeters (cm), but the length is in meters (m). We need to convert the length from meters to centimeters.

We know that 1 meter = 100 centimeters.

So, the length of the tube in centimeters is:

\(L = 2 \text{ m} \times 100 \text{ cm/m} = 200 \text{ cm}\)

Now all our dimensions are in centimeters: R = 4 cm, t = 0.5 cm, L = 200 cm.

Determining the Inner Radius

The outer radius (R) is the distance from the center to the outside surface of the tube. The thickness (t) is the material's depth. The inner radius (r) is the distance from the center to the inside surface of the tube. It is calculated by subtracting the thickness from the outer radius.

\(r = R - t\)

\(r = 4 \text{ cm} - 0.5 \text{ cm}\)

\(r = 3.5 \text{ cm}\)

Calculating the Volume of Metal

The metal used in the tube forms a hollow cylinder. The volume of a cylinder is given by the formula \(V = \pi \times (\text{radius})^2 \times \text{height}\) (or length in this case).

The volume of the metal is the difference between the volume of the cylinder defined by the outer radius and the volume of the cylinder defined by the inner radius, both having the same length L.

Volume of outer cylinder (including the hollow space) = \(\pi R^2 L\)

Volume of inner hollow space = \(\pi r^2 L\)

Volume of metal = Volume of outer cylinder - Volume of inner hollow space

Volume of metal = \(\pi R^2 L - \pi r^2 L\)

We can factor out \(\pi L\) from the expression:

Volume of metal = \(\pi L (R^2 - r^2)\)

Now, let's substitute the values we have:

• \(L = 200 \text{ cm}\)
• \(R = 4 \text{ cm}\)
• \(r = 3.5 \text{ cm}\)

Volume of metal = \(\pi \times 200 \times (4^2 - 3.5^2)\)

Calculate the squares:

• \(4^2 = 16\)
• \(3.5^2 = (7/2)^2 = 49/4 = 12.25\)

Calculate the difference in squares:

\(R^2 - r^2 = 16 - 12.25 = 3.75\)

Now, substitute this back into the volume formula:

Volume of metal = \(\pi \times 200 \times 3.75\)

Calculate the final product:

\(200 \times 3.75 = 200 \times (3 + 0.75) = 200 \times 3 + 200 \times 0.75 = 600 + 150 = 750\)

So, the volume of metal used is \(750 \pi\) cm³.

Final Answer Summary

The volume of metal used in making the cylindrical tube is \(750 \pi\) cubic centimeters.

Revision Table: Cylindrical Tube Volume Calculation

Additional Information: Hollow Cylinder Concepts

A hollow cylinder is essentially a larger cylinder with a smaller cylinder removed from its center. The volume of the material forming the hollow part is found by subtracting the volume of the inner void from the volume of the outer shape.

Key concepts involved:

• Cylinder Volume: The fundamental formula \(V = \pi r^2 h\) applies to the total volume enclosed by a cylinder's radius and height.
• Hollow Shape Volume: For hollow shapes like tubes, pipes, or rings (if thin), the volume of the material is the difference between the outer volume and the inner volume.
• Units Consistency: Always ensure all dimensions are in the same units before performing calculations to avoid errors.

The formula for the volume of material in a hollow cylinder with outer radius R, inner radius r, and length L is \(V = \pi (R^2 - r^2) L\). This can also be written as \(V = \pi (R-r)(R+r)L\). Since \(R-r\) is the thickness (t), the formula can also be expressed as \(V = \pi t (R+r) L\).

Let's verify using \(V = \pi t (R+r) L\):

• \(t = 0.5\) cm
• \(R + r = 4 + 3.5 = 7.5\) cm
• \(L = 200\) cm

\(V = \pi \times 0.5 \times 7.5 \times 200\)

\(V = \pi \times (0.5 \times 200) \times 7.5\)

\(V = \pi \times 100 \times 7.5\)

\(V = \pi \times 750 = 750 \pi\) cm³.

Both formulas yield the same result, confirming the calculation.