Problem 68 - Calculus

Question

An upright cylinder of radius \(r\) is sliced by two parallel planes at angle \(x\). One is a height \(h\) above the other. (a) Draw a picture to show that the volume between the planes is \(\pi r^{2} h .\) (b) Tilt the picture by \(x\), so the base and top are flat. What is the shape of the base? What is its area \(A\) ? What is the height \(H\) of the tilted cylinder?

Step-by-Step Solution

Answer

(a) Volume is \( \pi r^2 h \). (b) Base is an ellipse with area \( \pi r^2 / \cos(x) \), height is \( h / \cos(x) \).

Step 1: Understanding the Problem

We are asked to find the volume of a cylindrical section and visualize its orientation when tilted. The first task is to confirm the volume formula for a vertical slice, and the second requires considering geometric transformations.

Step 2: Visualizing the Cylinder Slice

Consider a cylinder with radius \( r \) and a height \( h \) between two parallel horizontal planes. The volume of this section between the planes is the volume of the cylinder with height \( h \), which is simply \( \pi r^2 h \). This is because a perpendicular section through a cylinder forms a right circular cylinder.

Step 3: Confirming the Volume Formula

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Since the planes are parallel and horizontal, the section maintains the shape of a cylinder, confirming that the volume is \( \pi r^2 h \).

Step 4: Tilt the Cylinder and Identify the New Shape

Tilting the cylinder effectively reorients it so that it rests with the bases flat on an incline. When considering the tilted perspective, the base appears as an ellipse, a result of slicing the circular base of the cylinder at an angle \( x \).

Step 5: Finding the Area of the Elliptical Base

The major axis of the ellipse aligns with the tilt, and the minor axis remains the original radius \( r \). The area of an ellipse is given by \( A = \pi a b \), where \( a \) is the semi-major axis length and \( b = r \). Since one dimension is unchanged by the tilting, \( a = r / \cos(x) \). Therefore, the elliptical area is \( A = \pi r^2 / \cos(x) \).

Step 6: Determining the Height of the Tilted Cylinder

The height \( H \) of the tilted cylinder is still the original vertical height \( h \), as the tilt doesn't change the actual vertical separation between the parallel planes. Thus, \( H = h / \cos(x) \).

Key Concepts

Cylinder Geometry

A cylinder is a 3-dimensional geometric figure with congruent circular bases connected by a curved surface. The simplicity of a cylinder's shape makes it easy to describe and calculate various properties, such as volume. Its geometry is characterized by two main measurements: the radius, denoted as \( r \), and the height, \( h \).
The volume of a cylinder is calculated using the formula:

• Volume (\( V \)) = \( \pi r^2 h \)

This formula is derived from the area of the base circle (\( \pi r^2 \)) times the height. Understanding this fundamental geometry helps solve more complex problems, like those involving transformations or changes in orientation.
When a cylinder is sliced by parallel planes, as described in the exercise, only the height \( h \) between the planes is relevant to finding the volume of that section. With the radius constant, the formula remains \( \pi r^2 h \) for the volume.

Ellipse Area

When a cylinder is tilted, its circular base appears as an ellipse when viewed from a different angle. An ellipse is a 2-dimensional shape characterized by two axes: the semi-major axis and the semi-minor axis. The area of an ellipse is calculated using:

• Area (\( A \)) = \( \pi a b \)

where \( a \) is the semi-major axis length, and \( b \) is the semi-minor axis length.
For the cylinder problem, when tilted at angle \( x \), the semi-minor axis remains the radius \( r \) of the original circle, while the semi-major axis extends due to the angle of tilt. Specifically, the elongated axis becomes \( r / \cos(x) \), resulting in the ellipse area:

• Ellipse Area \( A = \pi r^2 / \cos(x) \)

Understanding the nature of ellipses helps visualize geometric transformations and how shapes change under different perspectives.

Geometric Transformations

Geometric transformations involve changing the position or orientation of a shape without altering its basic properties. In this context, tilting a cylinder involves a rotational transformation around an axis. This type of transformation means the shape's characteristics, such as volume, remain constant despite visual appearance changes.
Tilting the cylinder affects how we perceive it, especially looking at its bases. The circular bases look elliptical from a non-perpendicular view. However, geometric transformations like rotation do not change distances between parallel planes, which means the height between them remains consistent.
These transformations are fundamental in geometry, allowing us to analyze and solve problems by viewing them from different angles and perspectives, understanding how shapes and their properties behave under various conditions.

Parallel Planes

Parallel planes are two planes that never intersect. In geometry, they maintain a constant distance from each other. Understanding this concept is crucial in the cylinder exercise, as the cylinder is sliced by two such planes creating a defined height \( h \) between them.
Even when the cylinder is tilted, these planes remain parallel, and the distance between them does not change. They affect how we compute the volume or transformations without affecting underlying geometric principles.

• The distance between parallel planes remains the fixed height of the section, \( h \).
• Tilting the cylinder alters the perception of the base but preserves the volume between these planes.

Knowing how parallel planes influence geometry allows us to understand and calculate sections of geometric figures accurately, irrespective of their orientation.