Calculating the Volume of Metal in a Hollow Cylinder

This question asks us to find the volume of metal used to create a hollow cylinder. A hollow cylinder is like a pipe; it has an outer boundary and an inner boundary, with the metal forming the region between them. To find the volume of the metal, we need to calculate the volume of the larger cylinder (including the hole) and subtract the volume of the inner, empty space (the hole).

Understanding the Dimensions of the Hollow Cylinder

We are given the following information about the hollow cylinder:

• Outer radius (\(R\)) = 4 cm
• Height (\(h\)) = 2 cm
• Thickness of the metal sheet = 1 cm
• We need to use \(\pi = \frac{22}{7}\) for calculations.

Determining the Inner Radius

The thickness of the metal sheet is the difference between the outer radius and the inner radius. If the outer radius is \(R\) and the thickness is \(t\), the inner radius (\(r\)) can be found using the formula:

\(r = R - t\)

Substituting the given values:

\(r = 4 \text{ cm} - 1 \text{ cm}\)

\(r = 3 \text{ cm}\)

So, the inner radius of the hollow cylinder is 3 cm.

Formula for the Volume of a Hollow Cylinder

The volume of a solid cylinder is given by the formula \(V = \pi r_{cylinder}^2 h\). For a hollow cylinder, the volume of the material used is the difference between the volume of the outer cylinder and the volume of the inner cylinder.

Volume of outer cylinder (\(V_{outer}\)) = \(\pi R^2 h\)

Volume of inner cylinder (\(V_{inner}\)) = \(\pi r^2 h\)

Volume of metal used (\(V_{metal}\)) = \(V_{outer} - V_{inner} = \pi R^2 h - \pi r^2 h\)

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

\(V_{metal} = \pi h (R^2 - r^2)\)

This is the standard formula for the volume of a hollow cylinder's material.

Step-by-Step Calculation of Metal Volume

Now, let's substitute the values we have into the formula for the volume of metal:

\(R = 4\) cm, \(r = 3\) cm, \(h = 2\) cm, and \(\pi = \frac{22}{7}\)

\(V_{metal} = \frac{22}{7} \times 2 \times (4^2 - 3^2)\)

First, calculate the squares of the radii:

\(4^2 = 4 \times 4 = 16\)

\(3^2 = 3 \times 3 = 9\)

Now, substitute these values back into the equation:

\(V_{metal} = \frac{22}{7} \times 2 \times (16 - 9)\)

Calculate the difference inside the parentheses:

\(16 - 9 = 7\)

Substitute this difference back:

\(V_{metal} = \frac{22}{7} \times 2 \times 7\)

Now, perform the multiplication. We can cancel out the 7 in the denominator with the 7 in the numerator:

\(V_{metal} = 22 \times 2 \times \frac{7}{7}\)

\(V_{metal} = 22 \times 2 \times 1\)

\(V_{metal} = 44\)

The volume of metal used is 44 cubic centimeters.

The unit for volume is cubic centimeters (cm\(^3\)) because the dimensions are given in centimeters.

Comparing with Given Options

Let's check the calculated volume against the provided options:

Our calculated volume is 44 cm\(^3\), which matches Option 3.

Revision Table: Hollow Cylinder Volume

Additional Information on Cylinder Calculations

Cylinders are important shapes in geometry and appear in many real-world applications. Understanding how to calculate their volume and surface area is crucial.

• Solid Cylinder Volume: The formula \(V = \pi r^2 h\) comes from multiplying the area of the circular base (\(\pi r^2\)) by the height (\(h\)).
• Hollow Cylinder Variations: Sometimes you might be given the inner radius and thickness, or the inner and outer diameters. Always convert to radii before using formulas. The formula \(V_{metal} = \pi h (R^2 - r^2)\) can also be written as \(V_{metal} = \pi h (R - r)(R + r)\), which might be easier for calculation in some cases. Note that \(R-r\) is the thickness.
• Surface Area: Calculating the surface area of a hollow cylinder is slightly more complex than a solid one. It involves the area of the outer curved surface (\(2\pi R h\)), the inner curved surface (\(2\pi r h\)), and the area of the two annular (ring-shaped) ends (\(2 \times (\pi R^2 - \pi r^2)\)). The total surface area is the sum of these parts: \(SA = 2\pi R h + 2\pi r h + 2\pi (R^2 - r^2)\).
• Units: Always pay attention to units. If dimensions are in centimeters, volume is in cubic centimeters (cm\(^3\)) and area is in square centimeters (cm\(^2\)).