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Chapter 2: Problem 7

The intersection of a plane with a right circular cylinder could be which of the following? I. A circle II. Parallel lines III. Intersecting lines (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

Short Answer

Expert verified
The correct answer is (D) I and II only.

Step by step solution

01

Analyze Intersection I

First, consider a plane intersecting a right circular cylinder such that it goes through the middle of the cylinder without touching the bases. In this case, the intersection is a circle. So, the statement I is correct.
02

Analyze Intersection II

Next, consider when a plane intersects the curved surface of the cylinder in such a manner that it never intersects with the base of the cylinder. This intersection creates two parallel lines. Therefore, statement II is also correct.
03

Analyze Intersection III

Finally, consider if it is possible for a plane to intersect a cylinder resulting in intersecting lines (a shape similar to the letter X or V). In case of a right circular cylinder, this is not possible because the plane can either intersect along a curve (creating a circle or ellipse), or along straight lines (parallel lines when the plane is not perpendicular to the base). Thus, statement III is incorrect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plane Intersections
When studying how planes interact with various 3D shapes, it's crucial to understand the concept of intersections. An intersection is where a plane cuts through a solid object, like a cylinder. Depending on the angle and position of the cut, the intersection can form different shapes.
  • If a plane slices through the center of the cylinder's curved surface without hitting the top or bottom base, it creates a circle. Imagine taking a thin slice straight through the middle of a round birthday cake!

  • If the plane intersects the curved surface while running parallel to the sides and never touches the top or bottom, it results in two parallel lines. Think of this like cutting a cylinder-shaped block of cheese lengthwise into strips.

  • However, if someone suggests intersecting lines (a cross or X shape), that's a misunderstanding when it comes to right circular cylinders. This kind of intersection isn't possible due to the cylinder鈥檚 symmetrical and straight edges.
Cylinder Properties
A cylinder is one of the most straightforward yet fascinating shapes in solid geometry. Understanding its unique properties can help us predict how planes intersect with it. Here are some essential insights about cylinders:
  • A cylinder has two identical and parallel circular bases, joined by a curved surface. This structure makes it symmetrical.

  • The height of a cylinder is the distance measured perpendicularly between the two circular bases.

  • The curved surface area, or lateral area, can be unfolded into a rectangle. This helps visualize the possible lines of intersection that a plane can create.
The simplicity of the cylinder shape aids in predicting the kinds of intersections that can occur - a circle, set of parallel lines, but not intersecting ones.
Geometric Shapes
Geometry is filled with diverse shapes, each with distinct properties and relations. When we think about geometric figures, we ponder how they interact in three-dimensional space, such as how a plane could intersect with a cylinder.
  • In solid geometry, shapes like cylinders, cones, and spheres are examined not just for their appearance, but for how they can integrate or divide space.

  • Understanding the fundamental geometric rules helps us predict outcomes, such as knowing which interactions create certain intersection types.

  • Solid geometry extends beyond simple planes and lines; it involves visualizing and manipulating four-dimensional space through projections and sections.
Studying these interactions, like those between planes and cylinders, offers insights into real-world applications, from architectural design to engineering challenges.

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Most popular questions from this chapter

Of the following lists of numbers, which has the largest standard deviation? (A) \(2,7,15\) (B) \(3,7,14\) (B) \(5,7,12\) (D) \(10,11,12\) (E) \(11,11,11\)

A cylinder has a base radius of 2 and a height of 9. To the nearest whole number, by how much does the lateral area exceed the sum of the areas of the two bases? (A) 101 (B) 96 (C) 88 (D) 81 (E) 75

How many distinguishable rearrangements of the letters in the word CONTEST start with the two vowels? (A) 120 (B) 60 (C) 10 (D) 10 (D) 5 (E) none of these

The volume of the region between two concentric spheres of radii 2 and 5 is (A) 28 (B) 66 (C) 113 (D) 368 (E) 490

A right circular cone whose base radius is 4 is inscribed in a sphere of radius 5 . What is the ratio of the volume of the cone to the volume of the sphere? (A) \(0.222 : 1\) (B) \(0.256 : 1\) (C) \(0.288 : 1\) (D) \(0.333 : 1\) (E) \(0.864 : 1\)
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