Problem 53 - University Calculus: Early Transcendentals

Question

Estimating volume Estimate the volume of material in a cylindrical shell with length 30 in., radius 6 in., and shell thickness 0.5 in. (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

The cylindrical shell's volume is approximately 541.65 cubic inches.

Step 1: Understand the Cylindrical Shell

A cylindrical shell can be visualized as a hollow cylinder with specific inner and outer radii and a certain length. The thickness given is the difference between the outer radius and the inner radius.

Step 2: Determine the Inner and Outer Radii

The radius given is the outer radius of the shell, which is 6 inches. The shell thickness is 0.5 inches. Thus, the inner radius is found by subtracting the thickness from the outer radius: \[5.5\, \text{inches} = 6\, \text{inches} - 0.5\, \text{inches}\]

Step 3: Calculate Inner and Outer Volumes

The volume of a cylinder is calculated using the formula \( V = \pi r^2 h \). First, find the outer volume using the outer radius (6 in):\[V_{\text{outer}} = \pi (6^2)(30)\]Then find the inner volume using the inner radius (5.5 in):\[V_{\text{inner}} = \pi (5.5^2)(30)\]

Step 4: Solve for Each Volume

Calculating the outer volume:\[V_{\text{outer}} = \pi (36)(30) = 1080\pi\, \text{cubic inches}\]Calculating the inner volume:\[V_{\text{inner}} = \pi (30.25)(30) = 907.5\pi\, \text{cubic inches}\]

Step 5: Calculate the Volume of the Shell

Subtract the inner volume from the outer volume to find the volume of the shell:\[V_{\text{shell}} = V_{\text{outer}} - V_{\text{inner}} = 1080\pi - 907.5\pi = 172.5\pi\, \text{cubic inches}\]

Step 6: Finalize the Calculation

To obtain the numerical value, multiply by \(\pi \approx 3.1416\):\[V_{\text{shell}} \approx 172.5 \times 3.1416 \approx 541.65\, \text{cubic inches}\]

Key Concepts

Volume of a Cylinder

The volume of a cylinder is a measure of how much material or "stuff" fits inside. Think about it as fulling a cylindrical container with water or sand.
The formula is quite simple: \[V = \pi r^2 h\]

• **\(V\)** is the volume.
• **\(\pi\)** is a special number, approximately 3.1416.
• **\(r\)** is the radius of the cylinder's base.
• **\(h\)** is the height or length of the cylinder.

To find the volume, you first square the radius, multiply it by \(\pi\), and finally by the height. This way, you can calculate how much space is inside any cylindrical shape.

Outer Radius

The outer radius of a cylindrical shell is an essential component when calculating its volume. It's the distance from the center to the outer edge. This measure helps outline the entire boundary of the external surface of the shell. In our example, the outer radius given is 6 inches. This means:

• Measure from the very center of the cylinder to one outer edge.
• Always consider the length of the cylinder, which, together with the radius, provides the surface area calculations needed for the volume.

An accurate outer radius is crucial as it directly influences the total volume and, consequently, the amount of material needed to create the shell.

Inner Radius

The inner radius is slightly different from the outer radius. It doesn't stretch to the outer edge of the shell but just within the start of the shell's material. To calculate the inner radius:

• Start with the outer radius, here it is 6 inches.
• Subtract the shell's thickness, which is 0.5 inches.
• Thus, the inner radius is 5.5 inches.

This is important because it represents the inside boundary, where no material exists, fully describing the hollow part of the shell, impacting both the internal and overall volume computation.

Shell Thickness

Shell thickness can be visualized as the breadth of the wall that forms the cylindrical shell. Given in the example at 0.5 inches, shell thickness helps in defining:

• **The distinction between inner and outer volumes:** This difference enables you to see how much space the material occupies.
• **Structural integrity:** The thicker the shell, the more robust it usually is.

In simple terms, when you know the outer and inner radius, the shell thickness is just this extra layer:\[\text{Shell Thickness} = \text{Outer Radius} - \text{Inner Radius}\]Understanding shell thickness is essential to solving and visualizing the volume problems of cylindrical shells.